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

Revision as of 18:47, 28 June 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 2^nd 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 2^nd 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 imaginarypart, 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.

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Madhurig, Vidhya

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

Revision as of 18:47, 28 June 2018

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@@ Line 38: / Line 38: @@
 '''Quadratic polynomials'''
-Let us find out more about a 2<sup>nd</sup> degree '''polynomial''' '''y =''' '''ax<sup>2</sup>+bx+c'''
+<sup>nd</sup> degree '''polynomial''' '''y =''' '''ax<sup>2</sup>+bx+c'''
 Parabola
-If '''a > 0''', parabola opens upwards, minimum '''vertex '''('''extremum''')
-If '''a < 0''', parabola opens downwards, maximum '''vertex'''
-|  | Let us find out more about a '''2<sup>nd</sup> degree polynomial'''.
-'''y equals a x squared plus b x plus c'''
-The '''function''' graphs as a parabola.
-If '''a''' is greater than 0, the parabola opens upwards and has a '''minimum vertex''' or '''extremum'''.
-If '''a''' is less than 0, it opens downwards and has a '''maximum vertex''' or '''extremum'''.
-|-
-|  | '''Slide Number 6'''
-'''Quadratic polynomials'''
 If parabola intersects '''x axis''', '''x intercepts''' are '''real roots'''.
-'''Real roots''' x = -b ± sqrt(b<sup>2</sup>-4ac)/2a
 If parabola does not intersect '''x axis''' at all, no '''real roots''', only '''complex'''
+|  | '''Quadratic polynomials'''
-Two types of '''roots''': '''real''' and '''complex'''
+Let us find out more about a '''2<sup>nd</sup> degree polynomial'''.
-|  | If the parabola intersects the''' x axis, '''the '''intercepts''' are real roots.
-'''Real roots''' are given by values of x.
+'''y equals a x squared plus b x plus c'''
-'''x''' is '''ratio''' '''of''' '''minus b plus or minus squareroot of b squared minus 4ac to 2a'''.
+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'''.
@@ Line 76: / Line 61: @@
 Let us look at '''complex''' numbers.
 |-
-|  | '''Slide Number 7'''
+|  | '''Slide Number 6'''
 '''Complex numbers, XY plane'''
 As we know,
-A '''complex number''' is expressed as '''''z'' = a + ''i''b''': 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}
-In the XY plane, '''a + ''i''b '''corresponds to the point ('''a, b''').
+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'''.
+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 ''i''b.'''
+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.
-‘'''''i''’''' is '''imaginary number''' and is equal to '''squareroot of minus 1'''.
+'''''i''''' is '''imaginary number''' and is equal to '''squareroot of minus 1'''.
-In the XY plane, '''a plus ''i''b '''corresponds to the point '''a comma b'''.
+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'''.
+In the '''complex plane''', '''x axis''' is called '''real axis''', '''y axis''' is called '''imaginary axis'''.
 |-
-|  | '''Slide Number 8'''
+|  | '''Slide Number 7'''
 '''Complex numbers, complex plane'''
@@ Line 120: / Line 105: @@
 According to''' Pythagoras’ theorem, ''r'' '''is equal to '''squareroot of a squared plus b squared.'''
-|-
-|  | '''Slide Number 9'''
-'''Complex numbers, complex plane'''
-[[Image:]]
-'''Argument ϴ''' = angle between''' real axis''' and line segment connecting '''''z''''' to O '''(0,0)''' in counter-clockwise direction
-'''Polar form''' of '''''z'' = a + ''i''b''' is
-'''''z'' = ''r'' (cosϴ + ''i'' sinϴ)'''
-where '''a= ''r''cosϴ, b=''r''sinϴ'''
-|  | '''Argument theta''' is angle between '''real axis''' and line segment connecting '''''z''''' to origin.
-It is in counter-clockwise direction.
-'''Polar form''' of '''''z''''' equals '''a plus ''i''b''' is
-'''''z'' '''equals''' ''r'' times cos theta plus ''i'' sin theta'''
-where '''a '''is equal to''' ''r'' cos theta '''and''' b is ''r'' sin theta'''
 |-
 |  | Show the '''GeoGebra''' window.
@@ Line 172: / Line 135: @@
 |  | Following the same steps, create '''sliders b''' and '''c'''.
 |  | Following the same steps, create '''sliders''' '''b''' and '''c'''.
-|-
-|  | Click on '''Slider '''tool >>''' '''click in''' Graphics view.'''
-|  | Click on''' Slider '''tool, click in '''Graphics''' view.
-|-
-|  | Type Name as '''b.'''
-|  | In the '''Name '''field of dialog box, type '''b'''.
-|-
-|  | Point to''' Min, Max '''and''' Increment '''values.
-|  | Set '''Min '''value as -5, '''Max '''value as 10 and Increment as 1.
-|-
-|  | Click '''OK''' button.
-|  | Click '''OK''' button.
-|-
-|  | Click on '''Slider '''tool >>''' '''click in''' Graphics view.'''
-|  | Again, click on''' Slider '''tool, click in '''Graphics''' view.
-|-
-|  | In '''Slider''' dialog-box, in the '''Name '''field, type '''c'''.
-|  | In the '''Name '''field of dialog box, type '''c'''.
-|-
-|  | Point to''' Min, Max '''and''' Increment '''values.
-|  | Set '''Min '''value as -5, '''Max '''value as 10 and Increment as 1.
-|-
-|  | Click '''OK''' button.
-|  | Click '''OK''' button.
 |-
 |  | In '''input bar''', type '''f(x):=a x^2+b x+c'''>>press '''Enter'''.
@@ Line 304: / Line 243: @@
 Drag column to adjust width.
-'''b squared minus 4ac '''is also called the''' determinant. '''
+'''b squared minus 4ac '''is also called the''' discriminant'''.
 |-
 |  | Type '''Root1''' and '''Root2''' in cells '''A4''' and '''A5'''>>press '''Enter'''.
@@ Line 323: / Line 262: @@
 |  | The value minus 15 appears in '''cell''' '''B1''' corresponding to '''b squared minus 4 a c''' for '''f x'''.
-Note: '''Determinant''' 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'''.
@@ Line 357: / Line 296: @@
 Again, a question mark appears in '''cell''' '''C5'''.
-There are no '''real''' solutions to the '''negative square root of the determinant'''.
+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 determinant 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. '''
 |  | 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 determinant to 2a'''.
+This should plot the '''root''' corresponding to '''ratio of minus b minus squareroot of discriminant to 2a'''.
 |-
 |  |
@@ Line 384: / Line 323: @@
 |-
 |  |
-|  | '''Determinant''' 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'''.
@@ Line 406: / Line 345: @@
 This '''complex root''' has '''real axis coordinate,''' '''minus b divided by 2a'''.
-Imaginary axis co-ordinate is '''squareroot of negative determinant divided by 2a'''.
+Imaginary axis co-ordinate is '''squareroot of negative discriminant divided by 2a'''.
 |-
 |  | Type '''(b10,c10)''' in the input bar>>press '''Enter.'''
@@ Line 413: / Line 352: @@
 This complex root has '''real axis coordinate, minus b divided by 2a. '''
-'''Imaginary axis''' co-ordinate is '''minus squareroot of negative determinant 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.
@@ Line 439: / Line 378: @@
 |  | Let us summarize.
 |-
-|  | '''Slide Number 10'''
+|  | '''Slide Number 8'''
 '''Summary'''
 |  | In this tutorial, we have learnt to:
@@ Line 446: / Line 385: @@
 Use a '''spreadsheet''' to calculate '''roots''' ('''real''' and '''complex''') for quadratic '''polynomials'''
 |-
-|  | '''Slide Number 11'''
+|  | '''Slide Number 9'''
 '''Assignment'''
 |  | As an assignment:
@@ Line 453: / Line 392: @@
 Calculate '''roots''' of the '''polynomials'''.
 |-
-|  | '''Slide Number 12'''
+|  | '''Slide Number 10'''
 '''About Spoken Tutorial project'''
 |  | The video at the following link summarizes the '''Spoken Tutorial''' project.
@@ Line 459: / Line 398: @@
 Please download and watch it.
 |-
-|  | '''Slide Number 13'''
+|  | '''Slide Number 11'''
 '''Spoken Tutorial workshops'''
 |  | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.
@@ Line 465: / Line 404: @@
 For more details, please write to us.
 |-
-|  | '''Slide Number 14'''
+|  | '''Slide Number 12'''
 '''Forum for specific questions:'''
 Do you have questions in THIS Spoken Tutorial?
@@ Line 479: / Line 418: @@
 |  | Please post your timed queries on this forum.
 |-
-|  | '''Slide Number 15'''
+|  | '''Slide Number 13'''
 '''Acknowledgement'''
 |  | '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.