Difference between revisions of "Applications-of-GeoGebra/C2/Complex-Roots-of-Quadratic-Equations/English"

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Line 4: Line 4:
  
 
|-
 
|-
−
| | '''Slide Number 1'''
+
|| '''Slide Number 1'''
  
 
'''Title Slide'''
 
'''Title Slide'''
−
| | Welcome to this tutorial on '''Complex Roots of Quadratic Equations'''
+
|| Welcome to this tutorial on '''Complex Roots of Quadratic Equations'''.
 
|-
 
|-
−
| | '''Slide Number 2'''
+
|| '''Slide Number 2'''
  
 
'''Learning Objectives'''
 
'''Learning Objectives'''
−
| | In this tutorial, we will learn to,
+
|| In this tutorial, we will learn to,
 
Plot graphs of''' '''quadratic '''functions'''
 
Plot graphs of''' '''quadratic '''functions'''
  
 
Calculate '''real''' and '''complex roots''' of quadratic '''functions'''
 
Calculate '''real''' and '''complex roots''' of quadratic '''functions'''
 
|-
 
|-
−
| | '''Slide Number 3'''
+
|| '''Slide Number 3'''
 +
 
 
'''Pre-requisites'''
 
'''Pre-requisites'''
−
| | To follow this tutorial, you should be familiar with:  
+
|| To follow this tutorial, you should be familiar with:  
 
'''GeoGebra''' interface
 
'''GeoGebra''' interface
  
Line 28: Line 29:
 
If not, for relevant tutorials, please visit our website.
 
If not, for relevant tutorials, please visit our website.
 
|-
 
|-
−
| | '''Slide Number 4'''
+
|| '''Slide Number 4'''
 +
 
 
'''System Requirement'''
 
'''System Requirement'''
−
| | Here I am using:
+
|| Here I am using:
 
'''Ubuntu Linux''' OS version 14.04
 
'''Ubuntu Linux''' OS version 14.04
  
 
'''Geogebra 5.0.388.0-d'''
 
'''Geogebra 5.0.388.0-d'''
 
|-
 
|-
−
| | '''Slide Number 5'''
+
|| '''Slide Number 5'''
 +
 
 
'''Quadratic polynomials'''
 
'''Quadratic polynomials'''
  
Line 45: Line 48:
  
 
If parabola does not intersect '''x axis''' at all, no '''real roots''', only '''complex'''
 
If parabola does not intersect '''x axis''' at all, no '''real roots''', only '''complex'''
−
| | '''Quadratic polynomials'''
+
|| '''Quadratic polynomials'''
  
 
Let us find out more about a '''2<sup>nd</sup> degree polynomial'''.  
 
Let us find out more about a '''2<sup>nd</sup> degree polynomial'''.  
Line 61: Line 64:
 
Let us look at '''complex''' numbers.  
 
Let us look at '''complex''' numbers.  
 
|-
 
|-
−
| | '''Slide Number 6'''
+
|| '''Slide Number 6'''
 +
 
 
'''Complex numbers, XY plane'''
 
'''Complex numbers, XY plane'''
  
 
As we know,
 
As we know,
  
−
A '''complex number''' is expressed as '''''z = a + b i''''': where '''''a''''' is the '''real''' part, '''''b i''''' is '''imaginary'''part, and '''a''' and '''b''' are constants.
+
A '''complex number''' is expressed as '''''z = a + b i''''': where '''''a''''' is the '''real''' part, '''''b i''''' is '''imaginary''' part, and '''a''' and '''b''' are constants.
  
 
'''Imaginary number, ''i'' '''= sqrt(-1}
 
'''Imaginary number, ''i'' '''= sqrt(-1}
Line 73: Line 77:
  
 
In the '''complex plane''', '''x axis''' is called '''real axis, y axis''' is called '''imaginary axis'''.
 
In the '''complex plane''', '''x axis''' is called '''real axis, y axis''' is called '''imaginary axis'''.
−
| | '''Complex numbers, XY plane'''
+
|| '''Complex numbers, XY plane'''
 
As we know,
 
As we know,
  
−
A '''complex number''' is expressed as '''''z''''' equals a plus b i'''''.
+
A '''complex number''' is expressed as '''''z''''' '''equals a plus''' '''''b i'''''.
  
 
'''''a''''' is the '''real''' part; '''''b i''''' is imaginary part<nowiki>; </nowiki>'''a''' and '''b''' are constants.
 
'''''a''''' is the '''real''' part; '''''b i''''' is imaginary part<nowiki>; </nowiki>'''a''' and '''b''' are constants.
Line 86: Line 90:
 
In the '''complex plane''', '''x axis''' is called '''real axis''', '''y axis''' is called '''imaginary axis'''.
 
In the '''complex plane''', '''x axis''' is called '''real axis''', '''y axis''' is called '''imaginary axis'''.
 
|-
 
|-
−
| | '''Slide Number 7'''
+
|| '''Slide Number 7'''
 
'''Complex numbers, complex plane'''
 
'''Complex numbers, complex plane'''
  
Line 96: Line 100:
  
 
'''''r'' = sqrt (a<sup>2</sup>+b<sup>2</sup>) (Pythagoras’ theorem)'''
 
'''''r'' = sqrt (a<sup>2</sup>+b<sup>2</sup>) (Pythagoras’ theorem)'''
−
| | '''Complex numbers, complex plane'''
+
|| '''Complex numbers, complex plane'''
  
 
In '''complex plane''', '''''z''''' is a '''vector'''.  
 
In '''complex plane''', '''''z''''' is a '''vector'''.  
Line 106: Line 110:
 
According to''' Pythagoras’ theorem, ''r'' '''is equal to '''squareroot of a squared plus b squared.'''
 
According to''' Pythagoras’ theorem, ''r'' '''is equal to '''squareroot of a squared plus b squared.'''
 
|-
 
|-
−
| | Show the '''GeoGebra''' window.
+
|| Show the '''GeoGebra''' window.
−
| | I have already opened '''GeoGebra''' interface.
+
|| I have already opened '''GeoGebra''' interface.
 
|-
 
|-
−
| | Click on '''Slider '''tool >>''' '''click in''' Graphics view.'''
+
|| Click on '''Slider '''tool >>''' '''click in''' Graphics view.'''
−
| | Click on''' Slider '''tool and then click in '''Graphics''' view.
+
|| Click on''' Slider '''tool and then click in '''Graphics''' view.
 
|-
 
|-
−
| | Point to the dialog box.
+
|| Point to the dialog box.
−
| | '''Slider''' dialog-box appears.
+
|| '''Slider''' dialog-box appears.
 
|-
 
|-
−
| | Point to '''Number''' radio button.
+
|| Point to '''Number''' radio button.
−
| | By default, '''Number''' radio-button is selected.
+
|| By default, '''Number''' radio-button is selected.
 
|-
 
|-
−
| | Type Name as '''a.'''
+
|| Type Name as '''a.'''
−
| | In the '''Name '''field, type '''a'''.
+
|| In the '''Name '''field, type '''a'''.
 
|-
 
|-
−
| | Point to''' Min, Max '''and''' Increment '''values.
+
|| Point to''' Min, Max '''and''' Increment '''values.
−
| | Set '''Min '''value as 1, '''Max '''value as 5 and Increment as 1.
+
|| Set '''Min '''value as 1, '''Max '''value as 5 and Increment as 1.
 
|-
 
|-
−
| | Click '''OK''' button.
+
|| Click '''OK''' button.
−
| | Click '''OK''' button.
+
|| Click '''OK''' button.
 
|-
 
|-
−
| | Point to the '''slider'''.
+
|| Point to the '''slider'''.
−
| | This creates a number '''slider''' named “'''a'''”.
+
|| This creates a number '''slider''' named “'''a'''”.
 
|-
 
|-
−
| | Drag to show the changing values.
+
|| Drag to show the changing values.
−
| | Using the '''slider''', '''a''' can have values from 1 to 5, in increments of 1.
+
|| Using the '''slider''', '''a''' can have values from 1 to 5, in increments of 1.
 
|-
 
|-
−
| | Following the same steps, create '''sliders b''' and '''c'''.  
+
|| Following the same steps, create '''sliders b''' and '''c'''.  
−
| | Following the same steps, create '''sliders''' '''b''' and '''c'''.  
+
|| Following the same steps, create '''sliders''' '''b''' and '''c'''.  
 
|-
 
|-
−
| | In '''input bar''', type '''f(x):=a x^2+b x+c'''>>press '''Enter'''.
+
|| In '''input bar''', type '''f(x):=a x^2+b x+c'''>>press '''Enter'''.
  
 
Drag boundary to see '''Algebra''' view properly.  
 
Drag boundary to see '''Algebra''' view properly.  
−
| | In '''input bar''', type the following line.
+
|| In '''input bar''', type the following line.
  
 
'''f x '''in parentheses '''colon equals a space x caret 2 plus b space x plus c'''.  
 
'''f x '''in parentheses '''colon equals a space x caret 2 plus b space x plus c'''.  
Line 149: Line 153:
 
Pay attention to the spaces''' '''indicating multiplication.  
 
Pay attention to the spaces''' '''indicating multiplication.  
 
|-
 
|-
−
| | Point to the equation for '''f(x)''' in '''Algebra view'''.
+
|| Point to the equation for '''f(x)''' in '''Algebra view'''.
−
| | Observe the equation for '''f of x''' in '''Algebra view'''.
+
|| Observe the equation for '''f of x''' in '''Algebra view'''.
 
|-
 
|-
−
| | On '''sliders''', move '''a''' to '''1''', '''b''' to '''-2''' and '''c''' to '''-3'''.
+
|| On '''sliders''', move '''a''' to '''1''', '''b''' to '''-2''' and '''c''' to '''-3'''.
−
| | Set '''slider''' '''a '''at '''1''', '''slider''' '''b''' at minus''' 2''' and '''slider''' '''c '''at minus''' 3'''.
+
|| Set '''slider''' '''a '''at '''1''', '''slider''' '''b''' at minus''' 2''' and '''slider''' '''c '''at minus''' 3'''.
 
|-
 
|-
−
| | Point to the equation '''f(x)=1x<sup>2</sup>-2x-3''' in '''Algebra view'''.  
+
|| Point to the equation '''f(x)=1x<sup>2</sup>-2x-3''' in '''Algebra view'''.  
−
| | The equation '''f of x equals 1 x squared minus 2 x minus 3''' appears in '''Algebra view'''.  
+
|| The equation '''f of x equals 1 x squared minus 2 x minus 3''' appears in '''Algebra view'''.  
 
|-
 
|-
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.  
+
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool.  
  
 
Click in '''Graphics''' view.  
 
Click in '''Graphics''' view.  
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.  
+
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool.  
  
 
Click in '''Graphics''' view.  
 
Click in '''Graphics''' view.  
 
|-
 
|-
−
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.  
+
|| Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.  
−
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.  
+
|| Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.  
 
|-
 
|-
−
| | Point to the parabola in '''Graphics View'''.  
+
|| Point to the parabola in '''Graphics View'''.  
−
| | '''Function''' '''f''' is a parabola, intersecting '''x axis''' at '''minus 1 comma 0''' and '''3 comma 0'''.  
+
|| '''Function''' '''f''' is a parabola, intersecting '''x axis''' at '''minus 1 comma 0''' and '''3 comma 0'''.  
  
 
Thus,''' root'''s of '''fx equals x squared minus 2x minus 3 '''are '''x equals minus 1''' and '''3'''.
 
Thus,''' root'''s of '''fx equals x squared minus 2x minus 3 '''are '''x equals minus 1''' and '''3'''.
 
|-
 
|-
−
| | Type '''Root(f)''' in input bar>>press '''Enter'''.
+
|| Type '''Root(f)''' in input bar>>press '''Enter'''.
−
| | In '''input bar''', type '''Root f '''in parentheses and press '''Enter'''.  
+
|| In '''input bar''', type '''Root f '''in parentheses and press '''Enter'''.  
 
|-
 
|-
−
| | Point to the '''roots''' in '''Algebra view''' and the '''intercepts''' in '''Graphics view.'''
+
|| Point to the '''roots''' in '''Algebra view''' and the '''intercepts''' in '''Graphics view.'''
−
| | The '''roots''' appear in '''Algebra view'''.  
+
|| The '''roots''' appear in '''Algebra view'''.  
  
 
They also appear as '''x-intercepts''' of the parabola in '''Graphics view'''.
 
They also appear as '''x-intercepts''' of the parabola in '''Graphics view'''.
 
|-
 
|-
−
| | Type '''Extremum(f)''' in Input bar>>press '''Enter'''.
+
|| Type '''Extremum(f)''' in Input bar>>press '''Enter'''.
−
| | In '''input bar''', type '''Extremum f''' in parentheses and press '''Enter'''.
+
|| In '''input bar''', type '''Extremum f''' in parentheses and press '''Enter'''.
 
|-
 
|-
−
| | Point to the '''extremum''' in the '''Algebra''' and '''Graphics views.'''
+
|| Point to the '''extremum''' in the '''Algebra''' and '''Graphics views.'''
−
| | The '''minimum vertex''' appears in '''Algebra''' and '''Graphics views'''.
+
|| The '''minimum vertex''' appears in '''Algebra''' and '''Graphics views'''.
 
|-
 
|-
−
| | Double click on point C ('''extremum''') in '''Graphics view'''>>Select '''Object Properties.'''
+
|| Double click on point C ('''extremum''') in '''Graphics view'''>>Select '''Object Properties.'''
−
| | After double clicking on point '''C''' in '''Graphics View''', select '''Object Properties.'''
+
|| After double clicking on point '''C''' in '''Graphics View''', select '''Object Properties.'''
 
|-
 
|-
−
| | Click on '''red''' color box.
+
|| Click on '''red''' color box.
−
| | From '''Color''' tab, change the color to red.
+
|| From '''Color''' tab, change the color to red.
 
|-
 
|-
−
| | Close the '''Preferences '''box.
+
|| Close the '''Preferences '''box.
−
| | Close the '''Preferences '''dialog-box.
+
|| Close the '''Preferences '''dialog-box.
 
|-
 
|-
−
| | Point to '''C''' in '''Algebra''' and '''Graphics''' views.
+
|| Point to '''C''' in '''Algebra''' and '''Graphics''' views.
−
| | '''Point C''' ('''extremum''' of '''f of x''') is red in '''Algebra''' and '''Graphics views'''.
+
|| '''Point C''' ('''extremum''' of '''f of x''') is red in '''Algebra''' and '''Graphics views'''.
 
|-
 
|-
−
| | Click on '''Move''' tool, drag '''a''' to '''1''', '''b''' to '''5''', '''c''' to '''10'''.
+
|| Click on '''Move''' tool, drag '''a''' to '''1''', '''b''' to '''5''', '''c''' to '''10'''.
−
| | Click on '''Move''' tool, set '''slider''' '''a''' at '''1''', '''slider''' '''b''' at '''5''', '''slider''' '''c''' at '''10'''.
+
|| Click on '''Move''' tool, set '''slider''' '''a''' at '''1''', '''slider''' '''b''' at '''5''', '''slider''' '''c''' at '''10'''.
 
|-
 
|-
−
| | Point to the equation '''f(x)=1x<sup>2</sup>+5x+10''' in '''Algebra view.'''  
+
|| Point to the equation '''f(x)=1x<sup>2</sup>+5x+10''' in '''Algebra view.'''  
−
| | The equation '''f of x equals 1 x squared plus 5x plus 10 '''appears in '''Algebra view'''.
+
|| The equation '''f of x equals 1 x squared plus 5x plus 10 '''appears in '''Algebra view'''.
 
|-
 
|-
−
| | Click in and drag '''Graphics''' view to see this parabola.  
+
|| Click in and drag '''Graphics''' view to see this parabola.  
−
| | Click in and drag '''Graphics''' view to see this parabola.  
+
|| Click in and drag '''Graphics''' view to see this parabola.  
 
|-
 
|-
−
| | Point to the parabola in '''Graphics View'''.  
+
|| Point to the parabola in '''Graphics View'''.  
−
| | It does not intersect the '''x-axis'''.
+
|| It does not intersect the '''x-axis'''.
 
|-
 
|-
−
| | Point to the '''roots''', '''points A and B''' in the '''Algebra view'''.  
+
|| Point to the '''roots''', '''points A and B''' in the '''Algebra view'''.  
−
| | '''Points A '''and''' B''' are undefined as the '''function''' does not intersect the '''x axis'''.
+
|| '''Points A '''and''' B''' are undefined as the '''function''' does not intersect the '''x axis'''.
 
|-
 
|-
−
| | Point to '''extremum''' point C in '''Algebra''' and '''Graphics views'''.  
+
|| Point to '''extremum''' point C in '''Algebra''' and '''Graphics views'''.  
−
| | '''Extremum''' (point '''C''') is shown in red in '''Algebra''' and '''Graphics views'''.  
+
|| '''Extremum''' (point '''C''') is shown in red in '''Algebra''' and '''Graphics views'''.  
 
|-
 
|-
−
| |  
+
||  
−
| | '''Function f of x equals x squared plus 5x plus 10 '''has no '''real roots'''.  
+
|| '''Function f of x equals x squared plus 5x plus 10 '''has no '''real roots'''.  
  
 
Let us see the '''complex root'''s of this equation.  
 
Let us see the '''complex root'''s of this equation.  
 
|-
 
|-
−
| | Click on '''View'''>>'''Spreadsheet'''.
+
|| Click on '''View'''>>'''Spreadsheet'''.
−
| | Click on '''View''', then on '''Spreadsheet'''.  
+
|| Click on '''View''', then on '''Spreadsheet'''.  
  
 
This opens a spreadsheet on the right side of the '''Graphics view'''.  
 
This opens a spreadsheet on the right side of the '''Graphics view'''.  
 
|-
 
|-
−
| | Click to close '''Algebra view'''.  
+
|| Click to close '''Algebra view'''.  
−
| | Click to close '''Algebra view'''.  
+
|| Click to close '''Algebra view'''.  
 
|-
 
|-
−
| | Drag the boundary to see '''Spreadsheet''' view properly.  
+
|| Drag the boundary to see '''Spreadsheet''' view properly.  
−
| | Drag the boundary to see '''Spreadsheet''' view properly.  
+
|| Drag the boundary to see '''Spreadsheet''' view properly.  
 
|-
 
|-
−
| | Point to the '''spreadsheet'''.  
+
|| Point to the '''spreadsheet'''.  
−
| | Type the following '''labels''' and formulae in the '''spreadsheet'''.  
+
|| Type the following '''labels''' and formulae in the '''spreadsheet'''.  
 
|-
 
|-
−
| | Type “'''b^2-4ac'''” in cell '''A1'''>>press '''Enter'''.
+
|| Type “'''b^2-4ac'''” in cell '''A1'''>>press '''Enter'''.
  
 
Drag column to adjust width.  
 
Drag column to adjust width.  
−
| | In '''cell A1''', type within quotes '''b caret 2 minus 4ac''' and press '''Enter.'''
+
|| In '''cell A1''', type within quotes '''b caret 2 minus 4ac''' and press '''Enter.'''
  
 
Drag column to adjust width.  
 
Drag column to adjust width.  
Line 245: Line 249:
 
'''b squared minus 4ac '''is also called the''' discriminant'''.
 
'''b squared minus 4ac '''is also called the''' discriminant'''.
 
|-
 
|-
−
| | Type '''Root1''' and '''Root2''' in cells '''A4''' and '''A5'''>>press '''Enter'''.
+
|| Type '''Root1''' and '''Root2''' in cells '''A4''' and '''A5'''>>press '''Enter'''.
−
| | In cells '''A4''' and '''A5''', type '''Root1''' and '''Root2, '''and press '''Enter.'''
+
|| In cells '''A4''' and '''A5''', type '''Root1''' and '''Root2, '''and press '''Enter.'''
 
|-
 
|-
−
| | Type '''Complex root1''' and '''Complex root2''' in '''A9''' and '''A10'''>>press '''Enter'''.
+
|| Type '''Complex root1''' and '''Complex root2''' in '''A9''' and '''A10'''>>press '''Enter'''.
−
| | In cells '''A9''' and '''A10, '''type '''Complex root1''' and '''Complex root2'''.  
+
|| In cells '''A9''' and '''A10, '''type '''Complex root1''' and '''Complex root2'''.  
  
 
Press '''Enter'''.  
 
Press '''Enter'''.  
 
|-
 
|-
−
| | Drag column to adjust width.  
+
|| Drag column to adjust width.  
−
| | Drag column to adjust width.  
+
|| Drag column to adjust width.  
 
|-
 
|-
−
| | Type '''b^2-4 a c''' in cell '''B1'''>>press '''Enter. '''
+
|| Type '''b^2-4 a c''' in cell '''B1'''>>press '''Enter. '''
−
| | In '''cell B1''', type '''b caret 2 minus 4 space a space c''' and press '''Enter. '''
+
|| In '''cell B1''', type '''b caret 2 minus 4 space a space c''' and press '''Enter. '''
 
|-
 
|-
−
| | Point to cell '''B1'''.  
+
|| Point to cell '''B1'''.  
−
| | The value minus 15 appears in '''cell''' '''B1''' corresponding to '''b squared minus 4 a c''' for '''f x'''.
+
|| The value minus 15 appears in '''cell''' '''B1''' corresponding to '''b squared minus 4 a c''' for '''f x'''.
  
 
Note: '''Discriminant''' is always negative for quadratic '''functions''' without '''real roots'''.  
 
Note: '''Discriminant''' is always negative for quadratic '''functions''' without '''real roots'''.  
 
|-
 
|-
−
| | Type “'''-b/2a'''” in cell '''B3'''>>press '''Enter'''.
+
|| Type “'''-b/2a'''” in cell '''B3'''>>press '''Enter'''.
−
| | In '''cell B3''', type within quotes '''minus b divided by 2a'''.  
+
|| In '''cell B3''', type within quotes '''minus b divided by 2a'''.  
  
 
Press '''Enter.'''
 
Press '''Enter.'''
 
|-
 
|-
−
| | Type '''–b/2 a''' in cell '''B4'''>>press '''Enter'''.
+
|| Type '''–b/2 a''' in cell '''B4'''>>press '''Enter'''.
−
| | In '''cell B4''', type '''minus b divided by 2 space a'''.  
+
|| In '''cell B4''', type '''minus b divided by 2 space a'''.  
  
 
Press '''Enter'''.  
 
Press '''Enter'''.  
Line 276: Line 280:
 
Note the value '''-2.5''' appear in cell '''B4'''.  
 
Note the value '''-2.5''' appear in cell '''B4'''.  
 
|-
 
|-
−
| | Type '''B4''' in cell '''B5'''>>press '''Enter'''.  
+
|| Type '''B4''' in cell '''B5'''>>press '''Enter'''.  
−
| | In '''cell B5''', type '''B4''' and press '''Enter. '''
+
|| In '''cell B5''', type '''B4''' and press '''Enter. '''
  
 
The value -'''2.5''' appears in cell '''B5 '''also.  
 
The value -'''2.5''' appears in cell '''B5 '''also.  
 
|-
 
|-
−
| | Type “'''+-sqrt(b^2-4ac)/2a'''” in cell '''C3'''>>press '''Enter.'''
+
|| Type “'''+-sqrt(b^2-4ac)/2a'''” in cell '''C3'''>>press '''Enter.'''
−
| | In '''cell C3''', type the following '''line''' and press '''Enter'''.  
+
|| In '''cell C3''', type the following '''line''' and press '''Enter'''.  
  
 
Within quotes, '''plus minus sqrt D divided by 2a'''
 
Within quotes, '''plus minus sqrt D divided by 2a'''
 
|-
 
|-
−
| | Type '''sqrt(B1)/2 a''' in cell '''C4'''>>press '''Enter.'''
+
|| Type '''sqrt(B1)/2 a''' in cell '''C4'''>>press '''Enter.'''
−
| | In '''cell C4''', type '''sqrt B1''' in parentheses''' divided by 2 space a''' and press '''Enter. '''
+
|| In '''cell C4''', type '''sqrt B1''' in parentheses''' divided by 2 space a''' and press '''Enter. '''
  
 
Note that a question mark appears in '''cell C4'''.  
 
Note that a question mark appears in '''cell C4'''.  
 
|-
 
|-
−
| | Type '''–C4''' in cell '''C5'''>>press '''Enter.'''
+
|| Type '''–C4''' in cell '''C5'''>>press '''Enter.'''
−
| | In '''cell C5''', type '''minus C4''' and press '''Enter. '''
+
|| In '''cell C5''', type '''minus C4''' and press '''Enter. '''
  
 
Again, a question mark appears in '''cell''' '''C5'''.  
 
Again, a question mark appears in '''cell''' '''C5'''.  
Line 298: Line 302:
 
There are no '''real''' solutions to the '''negative square root of the discriminant'''.  
 
There are no '''real''' solutions to the '''negative square root of the discriminant'''.  
 
|-
 
|-
−
| | Type '''(b4+c4,0)''' in the input bar>>press '''Enter.'''
+
|| Type '''(b4+c4,0)''' in the input bar>>press '''Enter.'''
−
| | In '''input bar''', type '''b4 plus c4 comma 0 in parentheses''' and press '''Enter.'''  
+
|| In '''input bar''', type '''b4 plus c4 comma 0 in parentheses''' and press '''Enter.'''  
  
 
This should '''plot''' the '''root''' corresponding to '''ratio of minus b plus squareroot of discriminant to 2a.'''
 
This should '''plot''' the '''root''' corresponding to '''ratio of minus b plus squareroot of discriminant to 2a.'''
 
|-
 
|-
−
| | Type '''(b5+c5,0)''' in the input bar>>press '''Enter. '''
+
|| Type '''(b5+c5,0)''' in the input bar>>press '''Enter. '''
−
| | In input bar, type '''b5 plus c5 comma 0''' '''in parentheses '''and press '''Enter. '''
+
|| In input bar, type '''b5 plus c5 comma 0''' '''in parentheses '''and press '''Enter. '''
  
 
This should plot the '''root''' corresponding to '''ratio of minus b minus squareroot of discriminant to 2a'''.
 
This should plot the '''root''' corresponding to '''ratio of minus b minus squareroot of discriminant to 2a'''.
 
|-
 
|-
−
| |  
+
||  
−
| | '''f x equals x squared plus 5x plus 10 '''has no '''real roots'''.  
+
|| '''f x equals x squared plus 5x plus 10 '''has no '''real roots'''.  
  
 
Hence, the points do not appear in '''Graphics view'''.  
 
Hence, the points do not appear in '''Graphics view'''.  
 
|-
 
|-
−
| | Click in and drag''' Graphics '''view to see this properly.''' '''
+
|| Click in and drag''' Graphics '''view to see this properly.''' '''
−
| | Click in and drag''' Graphics '''view to see this properly.''' '''
+
|| Click in and drag''' Graphics '''view to see this properly.''' '''
 
|-
 
|-
−
| | Type '''–b/2 a''' in cell '''B9'''>>press '''Enter'''.  
+
|| Type '''–b/2 a''' in cell '''B9'''>>press '''Enter'''.  
−
| | In '''cell B9''', type '''minus b divided by 2 space a '''and press '''Enter.'''
+
|| In '''cell B9''', type '''minus b divided by 2 space a '''and press '''Enter.'''
 
|-
 
|-
−
| | In '''cell B10''',''' '''type''' B9 '''and press''' Enter.'''
+
|| In '''cell B10''',''' '''type''' B9 '''and press''' Enter.'''
−
| | In '''cell B10''',''' '''type''' B9 '''and press''' Enter.'''
+
|| In '''cell B10''',''' '''type''' B9 '''and press''' Enter.'''
 
|-
 
|-
−
| |  
+
||  
−
| | '''Discriminant''' is less than 0 for '''f x equals x squared plus 5x plus 10'''.  
+
|| '''Discriminant''' is less than 0 for '''f x equals x squared plus 5x plus 10'''.  
  
 
So the opposite sign will be taken to allow calculation of '''roots'''.  
 
So the opposite sign will be taken to allow calculation of '''roots'''.  
 
|-
 
|-
−
| | Type '''sqrt(-B1)/2 a''' in cell '''C9'''>>press '''Enter.'''
+
|| Type '''sqrt(-B1)/2 a''' in cell '''C9'''>>press '''Enter.'''
−
| | In '''cell C9''', type '''sqrt minus B1''' in parentheses''' divided by 2 space a''' and press '''Enter.'''
+
|| In '''cell C9''', type '''sqrt minus B1''' in parentheses''' divided by 2 space a''' and press '''Enter.'''
  
 
1.94 appears in '''C9'''.  
 
1.94 appears in '''C9'''.  
 
|-
 
|-
−
| | Type '''–C9''' in cell '''C10'''>>press '''Enter.'''
+
|| Type '''–C9''' in cell '''C10'''>>press '''Enter.'''
−
| | In '''cell C10''', type '''minus C9''' and press '''Enter.'''
+
|| In '''cell C10''', type '''minus C9''' and press '''Enter.'''
  
 
'''Minus''' 1.94 appears in '''C10'''.  
 
'''Minus''' 1.94 appears in '''C10'''.  
 
|-
 
|-
−
| | Click in and drag '''Graphics''' view to see both '''roots'''.
+
|| Click in and drag '''Graphics''' view to see both '''roots'''.
−
| | Click in and drag '''Graphics''' view to see the following '''complex''' '''roots'''.  
+
|| Click in and drag '''Graphics''' view to see the following '''complex''' '''roots'''.  
 
|-
 
|-
−
| | Type '''(b9,c9)''' in the input bar>>press '''Enter. '''
+
|| Type '''(b9,c9)''' in the input bar>>press '''Enter. '''
−
| | In '''input bar''', type '''b9 comma c9''' in parentheses and press '''Enter. '''
+
|| In '''input bar''', type '''b9 comma c9''' in parentheses and press '''Enter. '''
  
 
This '''complex root''' has '''real axis coordinate,''' '''minus b divided by 2a'''.  
 
This '''complex root''' has '''real axis coordinate,''' '''minus b divided by 2a'''.  
Line 347: Line 351:
 
Imaginary axis co-ordinate is '''squareroot of negative discriminant divided by 2a'''.  
 
Imaginary axis co-ordinate is '''squareroot of negative discriminant divided by 2a'''.  
 
|-
 
|-
−
| | Type '''(b10,c10)''' in the input bar>>press '''Enter.'''
+
|| Type '''(b10,c10)''' in the input bar>>press '''Enter.'''
−
| | In input bar, type '''b10 comma''' '''c10 in parentheses''' and press '''Enter. '''
+
|| In input bar, type '''b10 comma''' '''c10 in parentheses''' and press '''Enter. '''
  
 
This complex root has '''real axis coordinate, minus b divided by 2a. '''
 
This complex root has '''real axis coordinate, minus b divided by 2a. '''
Line 354: Line 358:
 
'''Imaginary axis''' co-ordinate is '''minus squareroot of negative discriminant divided by 2a'''.
 
'''Imaginary axis''' co-ordinate is '''minus squareroot of negative discriminant divided by 2a'''.
 
|-
 
|-
−
| | Drag boundary to see '''sliders''' in '''Graphics''' view properly.  
+
|| Drag boundary to see '''sliders''' in '''Graphics''' view properly.  
−
| | Drag boundary to see '''sliders''' in '''Graphics''' view properly.  
+
|| Drag boundary to see '''sliders''' in '''Graphics''' view properly.  
 
|-
 
|-
−
| | Drag the '''slider''' '''b''' to -2 and '''c''' to -3.
+
|| Drag the '''slider''' '''b''' to -2 and '''c''' to -3.
−
| | Drag the '''slider''' '''b''' to -2 and '''slider''' '''c''' to -3.
+
|| Drag the '''slider''' '''b''' to -2 and '''slider''' '''c''' to -3.
 
|-
 
|-
−
| | Click in and drag '''Graphics''' view to see the parabola.  
+
|| Click in and drag '''Graphics''' view to see the parabola.  
−
| | Click in and drag '''Graphics''' view to see the parabola.  
+
|| Click in and drag '''Graphics''' view to see the parabola.  
 
|-
 
|-
−
| | Point to the parabola in '''Graphics view'''.
+
|| Point to the parabola in '''Graphics view'''.
−
| | Note how the parabola changes to the one seen for '''f x equals x squared minus 2x minus 3'''.
+
|| Note how the parabola changes to the one seen for '''f x equals x squared minus 2x minus 3'''.
 
|-
 
|-
−
| | Point to the '''roots''' appearing at x '''intercepts''' of parabola in '''Graphics view'''.
+
|| Point to the '''roots''' appearing at x '''intercepts''' of parabola in '''Graphics view'''.
−
| | The '''real roots''' plotted earlier for '''f x equals x squared minus 2x minus 3''' appear now.  
+
|| The '''real roots''' plotted earlier for '''f x equals x squared minus 2x minus 3''' appear now.  
 
|-
 
|-
−
| | Drag boundary to see '''Spreadsheet''' view.  
+
|| Drag boundary to see '''Spreadsheet''' view.  
−
| | Drag boundary to see '''Spreadsheet''' view.  
+
|| Drag boundary to see '''Spreadsheet''' view.  
 
|-
 
|-
−
| | Point to the question marks appearing in '''C9''' and '''C10''' in the '''spreadsheet'''.  
+
|| Point to the question marks appearing in '''C9''' and '''C10''' in the '''spreadsheet'''.  
−
| | As '''roots''' are '''real''', calculations for '''complex roots''' become invalid.  
+
|| As '''roots''' are '''real''', calculations for '''complex roots''' become invalid.  
 
|-
 
|-
−
| |  
+
||  
−
| | Let us summarize.
+
|| Let us summarize.
 
|-
 
|-
−
| | '''Slide Number 8'''
+
|| '''Slide Number 8'''
 
'''Summary'''
 
'''Summary'''
−
| | In this tutorial, we have learnt to:
+
|| In this tutorial, we have learnt to:
 
Visualize quadratic '''polynomials''', their '''roots''' and '''extrema'''
 
Visualize quadratic '''polynomials''', their '''roots''' and '''extrema'''
  
 
Use a '''spreadsheet''' to calculate '''roots''' ('''real''' and '''complex''') for quadratic '''polynomials'''  
 
Use a '''spreadsheet''' to calculate '''roots''' ('''real''' and '''complex''') for quadratic '''polynomials'''  
 
|-
 
|-
−
| | '''Slide Number 9'''
+
|| '''Slide Number 9'''
 
'''Assignment'''
 
'''Assignment'''
−
| | As an assignment:
+
|| As an assignment:
 
Drag '''sliders''' to graph different quadratic '''polynomials'''.
 
Drag '''sliders''' to graph different quadratic '''polynomials'''.
  
 
Calculate '''roots''' of the '''polynomials'''.
 
Calculate '''roots''' of the '''polynomials'''.
 
|-
 
|-
−
| | '''Slide Number 10'''
+
|| '''Slide Number 10'''
 
'''About Spoken Tutorial project'''
 
'''About Spoken Tutorial project'''
−
| | The video at the following link summarizes the '''Spoken Tutorial''' project.
+
|| The video at the following link summarizes the '''Spoken Tutorial''' project.
  
 
Please download and watch it.
 
Please download and watch it.
 
|-
 
|-
−
| | '''Slide Number 11'''
+
|| '''Slide Number 11'''
 
'''Spoken Tutorial workshops'''
 
'''Spoken Tutorial workshops'''
−
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.
+
|| The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.
  
 
For more details, please write to us.
 
For more details, please write to us.
 
|-
 
|-
−
| | '''Slide Number 12'''
+
|| '''Slide Number 12'''
 
'''Forum for specific questions:'''
 
'''Forum for specific questions:'''
 
Do you have questions in THIS Spoken Tutorial?
 
Do you have questions in THIS Spoken Tutorial?
Line 416: Line 420:
 
Someone from our team will answer them.
 
Someone from our team will answer them.
  
−
| | Please post your timed queries on this forum.
+
|| Please post your timed queries on this forum.
 
|-
 
|-
−
| | '''Slide Number 13'''
+
|| '''Slide Number 13'''
 
'''Acknowledgement'''
 
'''Acknowledgement'''
−
| | '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.
+
|| '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.
  
 
More information on this mission is available at this link.
 
More information on this mission is available at this link.
 
|-
 
|-
−
| |  
+
||  
−
| | This is Vidhya Iyer from '''IIT Bombay''', signing off.
+
|| This is Vidhya Iyer from '''IIT Bombay''', signing off.
  
 
Thank you for joining.
 
Thank you for joining.
 
|-
 
|-
 
|}
 
|}

Revision as of 13:28, 4 July 2018

Visual Cue Narration
Slide Number 1

Title Slide

Welcome to this tutorial on Complex Roots of Quadratic Equations.
Slide Number 2

Learning Objectives

In this tutorial, we will learn to,

Plot graphs of quadratic functions

Calculate real and complex roots of quadratic functions

Slide Number 3

Pre-requisites

To follow this tutorial, you should be familiar with:

GeoGebra interface

Basics of quadratic equations, geometry and graphs

Previous tutorials in this series

If not, for relevant tutorials, please visit our website.

Slide Number 4

System Requirement

Here I am using:

Ubuntu Linux OS version 14.04

Geogebra 5.0.388.0-d

Slide Number 5

Quadratic polynomials

2nd degree polynomial y = ax2+bx+c

Parabola

If parabola intersects x axis, x intercepts are real roots.

If parabola does not intersect x axis at all, no real roots, only complex

Quadratic polynomials

Let us find out more about a 2nd degree polynomial.

y equals a x squared plus b x plus c

The function graphs as a parabola.

If the parabola intersects the x axis, the intercepts are real roots.

If the parabola does not intersect x axis at all, it has no real roots.

Roots are complex.

Let us look at complex numbers.

Slide Number 6

Complex numbers, XY plane

As we know,

A complex number is expressed as z = a + b i: where a is the real part, b i is imaginary part, and a and b are constants.

Imaginary number, i = sqrt(-1}

In the XY plane, a + b i corresponds to the point (a, b).

In the complex plane, x axis is called real axis, y axis is called imaginary axis.

Complex numbers, XY plane

As we know,

A complex number is expressed as z equals a plus b i.

a is the real part; b i is imaginary part; a and b are constants.

i is imaginary number and is equal to squareroot of minus 1.

In the XY plane, a plus b i corresponds to the point a comma b.

In the complex plane, x axis is called real axis, y axis is called imaginary axis.

Slide Number 7

Complex numbers, complex plane

[[Image:]]

In complex plane, z is a vector with real axis coordinate ‘a’ and imaginary axis coordinate ‘b’

Length of the vector z = |z| = r

r = sqrt (a2+b2) (Pythagoras’ theorem)

Complex numbers, complex plane

In complex plane, z is a vector.

Its real axis coordinate is ‘a’ and imaginary axis coordinate is ‘b’.

The length of the vector z is equal to the absolute value of z and to r.

According to Pythagoras’ theorem, r is equal to squareroot of a squared plus b squared.

Show the GeoGebra window. I have already opened GeoGebra interface.
Click on Slider tool >> click in Graphics view. Click on Slider tool and then click in Graphics view.
Point to the dialog box. Slider dialog-box appears.
Point to Number radio button. By default, Number radio-button is selected.
Type Name as a. In the Name field, type a.
Point to Min, Max and Increment values. Set Min value as 1, Max value as 5 and Increment as 1.
Click OK button. Click OK button.
Point to the slider. This creates a number slider named “a”.
Drag to show the changing values. Using the slider, a can have values from 1 to 5, in increments of 1.
Following the same steps, create sliders b and c. Following the same steps, create sliders b and c.
In input bar, type f(x):=a x^2+b x+c>>press Enter.

Drag boundary to see Algebra view properly.

In input bar, type the following line.

f x in parentheses colon equals a space x caret 2 plus b space x plus c.

Press Enter.

Drag boundary to see Algebra view properly.

Pay attention to the spaces indicating multiplication.

Point to the equation for f(x) in Algebra view. Observe the equation for f of x in Algebra view.
On sliders, move a to 1, b to -2 and c to -3. Set slider a at 1, slider b at minus 2 and slider c at minus 3.
Point to the equation f(x)=1x2-2x-3 in Algebra view. The equation f of x equals 1 x squared minus 2 x minus 3 appears in Algebra view.
Under Move Graphics View, click on Zoom Out tool.

Click in Graphics view.

Under Move Graphics View, click on Zoom Out tool.

Click in Graphics view.

Click on Move Graphics View tool and drag Graphics view to see parabola f. Click on Move Graphics View tool and drag Graphics view to see parabola f.
Point to the parabola in Graphics View. Function f is a parabola, intersecting x axis at minus 1 comma 0 and 3 comma 0.

Thus, roots of fx equals x squared minus 2x minus 3 are x equals minus 1 and 3.

Type Root(f) in input bar>>press Enter. In input bar, type Root f in parentheses and press Enter.
Point to the roots in Algebra view and the intercepts in Graphics view. The roots appear in Algebra view.

They also appear as x-intercepts of the parabola in Graphics view.

Type Extremum(f) in Input bar>>press Enter. In input bar, type Extremum f in parentheses and press Enter.
Point to the extremum in the Algebra and Graphics views. The minimum vertex appears in Algebra and Graphics views.
Double click on point C (extremum) in Graphics view>>Select Object Properties. After double clicking on point C in Graphics View, select Object Properties.
Click on red color box. From Color tab, change the color to red.
Close the Preferences box. Close the Preferences dialog-box.
Point to C in Algebra and Graphics views. Point C (extremum of f of x) is red in Algebra and Graphics views.
Click on Move tool, drag a to 1, b to 5, c to 10. Click on Move tool, set slider a at 1, slider b at 5, slider c at 10.
Point to the equation f(x)=1x2+5x+10 in Algebra view. The equation f of x equals 1 x squared plus 5x plus 10 appears in Algebra view.
Click in and drag Graphics view to see this parabola. Click in and drag Graphics view to see this parabola.
Point to the parabola in Graphics View. It does not intersect the x-axis.
Point to the roots, points A and B in the Algebra view. Points A and B are undefined as the function does not intersect the x axis.
Point to extremum point C in Algebra and Graphics views. Extremum (point C) is shown in red in Algebra and Graphics views.
Function f of x equals x squared plus 5x plus 10 has no real roots.

Let us see the complex roots of this equation.

Click on View>>Spreadsheet. Click on View, then on Spreadsheet.

This opens a spreadsheet on the right side of the Graphics view.

Click to close Algebra view. Click to close Algebra view.
Drag the boundary to see Spreadsheet view properly. Drag the boundary to see Spreadsheet view properly.
Point to the spreadsheet. Type the following labels and formulae in the spreadsheet.
Type “b^2-4ac” in cell A1>>press Enter.

Drag column to adjust width.

In cell A1, type within quotes b caret 2 minus 4ac and press Enter.

Drag column to adjust width.

b squared minus 4ac is also called the discriminant.

Type Root1 and Root2 in cells A4 and A5>>press Enter. In cells A4 and A5, type Root1 and Root2, and press Enter.
Type Complex root1 and Complex root2 in A9 and A10>>press Enter. In cells A9 and A10, type Complex root1 and Complex root2.

Press Enter.

Drag column to adjust width. Drag column to adjust width.
Type b^2-4 a c in cell B1>>press Enter. In cell B1, type b caret 2 minus 4 space a space c and press Enter.
Point to cell B1. The value minus 15 appears in cell B1 corresponding to b squared minus 4 a c for f x.

Note: Discriminant is always negative for quadratic functions without real roots.

Type “-b/2a” in cell B3>>press Enter. In cell B3, type within quotes minus b divided by 2a.

Press Enter.

Type –b/2 a in cell B4>>press Enter. In cell B4, type minus b divided by 2 space a.

Press Enter.

Note the value -2.5 appear in cell B4.

Type B4 in cell B5>>press Enter. In cell B5, type B4 and press Enter.

The value -2.5 appears in cell B5 also.

Type “+-sqrt(b^2-4ac)/2a” in cell C3>>press Enter. In cell C3, type the following line and press Enter.

Within quotes, plus minus sqrt D divided by 2a

Type sqrt(B1)/2 a in cell C4>>press Enter. In cell C4, type sqrt B1 in parentheses divided by 2 space a and press Enter.

Note that a question mark appears in cell C4.

Type –C4 in cell C5>>press Enter. In cell C5, type minus C4 and press Enter.

Again, a question mark appears in cell C5.

There are no real solutions to the negative square root of the discriminant.

Type (b4+c4,0) in the input bar>>press Enter. In input bar, type b4 plus c4 comma 0 in parentheses and press Enter.

This should plot the root corresponding to ratio of minus b plus squareroot of discriminant to 2a.

Type (b5+c5,0) in the input bar>>press Enter. In input bar, type b5 plus c5 comma 0 in parentheses and press Enter.

This should plot the root corresponding to ratio of minus b minus squareroot of discriminant to 2a.

f x equals x squared plus 5x plus 10 has no real roots.

Hence, the points do not appear in Graphics view.

Click in and drag Graphics view to see this properly. Click in and drag Graphics view to see this properly.
Type –b/2 a in cell B9>>press Enter. In cell B9, type minus b divided by 2 space a and press Enter.
In cell B10, type B9 and press Enter. In cell B10, type B9 and press Enter.
Discriminant is less than 0 for f x equals x squared plus 5x plus 10.

So the opposite sign will be taken to allow calculation of roots.

Type sqrt(-B1)/2 a in cell C9>>press Enter. In cell C9, type sqrt minus B1 in parentheses divided by 2 space a and press Enter.

1.94 appears in C9.

Type –C9 in cell C10>>press Enter. In cell C10, type minus C9 and press Enter.

Minus 1.94 appears in C10.

Click in and drag Graphics view to see both roots. Click in and drag Graphics view to see the following complex roots.
Type (b9,c9) in the input bar>>press Enter. In input bar, type b9 comma c9 in parentheses and press Enter.

This complex root has real axis coordinate, minus b divided by 2a.

Imaginary axis co-ordinate is squareroot of negative discriminant divided by 2a.

Type (b10,c10) in the input bar>>press Enter. In input bar, type b10 comma c10 in parentheses and press Enter.

This complex root has real axis coordinate, minus b divided by 2a.

Imaginary axis co-ordinate is minus squareroot of negative discriminant divided by 2a.

Drag boundary to see sliders in Graphics view properly. Drag boundary to see sliders in Graphics view properly.
Drag the slider b to -2 and c to -3. Drag the slider b to -2 and slider c to -3.
Click in and drag Graphics view to see the parabola. Click in and drag Graphics view to see the parabola.
Point to the parabola in Graphics view. Note how the parabola changes to the one seen for f x equals x squared minus 2x minus 3.
Point to the roots appearing at x intercepts of parabola in Graphics view. The real roots plotted earlier for f x equals x squared minus 2x minus 3 appear now.
Drag boundary to see Spreadsheet view. Drag boundary to see Spreadsheet view.
Point to the question marks appearing in C9 and C10 in the spreadsheet. As roots are real, calculations for complex roots become invalid.
Let us summarize.
Slide Number 8

Summary

In this tutorial, we have learnt to:

Visualize quadratic polynomials, their roots and extrema

Use a spreadsheet to calculate roots (real and complex) for quadratic polynomials

Slide Number 9

Assignment

As an assignment:

Drag sliders to graph different quadratic polynomials.

Calculate roots of the polynomials.

Slide Number 10

About Spoken Tutorial project

The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.

Slide Number 11

Spoken Tutorial workshops

The Spoken Tutorial Project team conducts workshops and gives certificates.

For more details, please write to us.

Slide Number 12

Forum for specific questions: Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

Please post your timed queries on this forum.
Slide Number 13

Acknowledgement

Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Madhurig, Vidhya

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