Applications-of-GeoGebra/C2/Complex-Roots-of-Quadratic-Equations/English - Revision history

Madhurig at 07:15, 5 July 2018

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Madhurig at 07:07, 5 July 2018

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Madhurig at 10:09, 4 July 2018

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Madhurig at 09:52, 4 July 2018

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Madhurig at 08:12, 4 July 2018

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Madhurig at 07:58, 4 July 2018

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Vidhya at 13:17, 28 June 2018

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Vidhya: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Complex Roots of Quadratic Equations''' |-..."

2018-03-16T08:55:20Z

Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Complex Roots of Quadratic Equations''' |-..."

New page

{|border=1
||'''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'''

Let us find out more about a 2nd degree '''polynomial''' '''y =''' '''ax2+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 '''2nd 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(b2-4ac)/2a

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

Two types of '''roots''': '''real''' and '''complex'''
| | If the parabola intersects the''' x axis, '''the '''intercepts''' are real roots.

'''Real roots''' are given by values of x.

'''x''' is '''ratio''' '''of''' '''minus b plus or minus squareroot of b squared minus 4ac to 2a'''.

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 7'''
'''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.

'''Imaginary number, ''i'' '''= sqrt(-1}

In the XY plane, '''a + ''i''b '''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 ''i''b.'''

‘'''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'''.

In the XY plane, '''a plus ''i''b '''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 8'''
'''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.'''
|-
| | '''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.
| | 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'''.
|-
| | 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'''.

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,''' 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'''.
| | 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 root'''s 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''' determinant. '''
|-
| | 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: '''Determinant''' 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 determinant'''.
|-
| | 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.'''
|-
| | 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'''.
|-
| |
| | '''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.'''
|-
| |
| | '''Determinant''' 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 determinant 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 determinant 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 10'''
'''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 11'''
'''Assignment'''
| | As an assignment:
Drag '''sliders''' to graph different quadratic '''polynomials'''.

Calculate '''roots''' of the '''polynomials'''.
|-
| | '''Slide Number 12'''
'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 13'''
'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 14'''
'''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 15'''
'''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.
|-
|}

@@ Line 303: / Line 303: @@
 |-
 || 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 square root of discriminant to 2a.
@@ Line 313: / Line 313: @@
 |-
 || Point to the graph.
-|| '''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'''.
@@ Line 347: / Line 347: @@
 || 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.
 Imaginary axis co-ordinate is square root of negative '''discriminant''' divided by 2a.

@@ Line 215: / Line 215: @@
 || 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'''.
 |-
@@ Line 240: / Line 240: @@
 || 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.
@@ Line 250: / Line 250: @@
 |-
 || 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'''.
@@ Line 260: / Line 260: @@
 || 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'''.
 |-
@@ Line 313: / Line 313: @@
 |-
 || Point to the graph.
-|| '''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'''.
@@ Line 332: / Line 332: @@
 |-
 || 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'''.
 .94 appears in '''C9'''.
@@ Line 361: / Line 361: @@
 || 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 >> '''c''' to -3.
 || Drag the '''slider b''' to -2 and '''slider c''' to -3.
 |-