PoojaMoolya: Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- || 00:01 || Welcome to this tutorial on '''Complex Roots of Quadratic Equations'''. |- || 00:07 || In this tutorial, we will lea..."

2020-10-21T07:23:31Z

Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- || 00:01 || Welcome to this tutorial on '''Complex Roots of Quadratic Equations'''. |- || 00:07 || In this tutorial, we will lea..."

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{|border=1
||'''Time'''
||'''Narration'''

|-
|| 00:01
|| Welcome to this tutorial on '''Complex Roots of Quadratic Equations'''.

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|| 00:07
|| In this tutorial, we will learn to

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|| 00:10
|| Plot graphs of '''quadratic '''functions'''

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|| 00:14
|| Calculate '''real''' and '''complex roots''' of quadratic functions.

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|| 00:19
|| To follow this tutorial, you should be familiar with:

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|| 00:22
|| '''GeoGebra''' interface

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|| 00:25
|| Basics of quadratic equations, geometry and graphs

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|| 00:30
|| Previous tutorials in this series

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|| 00:33
|| If not, for relevant tutorials, please visit our website.
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||00:38
|| Here I am using:

'''Ubuntu Linux''' OS version 14.04, '''Geogebra 5.0.388.0 hyphen d'''
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|| 00:53
|| '''Quadratic polynomials'''

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|| 00:56
|| Let us find out more about a '''2<sup>nd</sup>''' degree polynomial.

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|| 01:01
|| y equals a x squared plus b x plus c

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|| 01:06
|| The '''function''' graphs as a parabola.

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|| 01:09
|| If the parabola intersects the x axis, the '''intercepts''' are real roots.

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|| 01:15
|| If the parabola does not intersect x axis at all, it has no '''real roots'''.

'''Roots''' are '''complex'''.

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|| 01:24
|| Let us look at '''complex''' numbers.
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|| 01:27
|| '''Complex numbers, XY plane'''

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|| 01:30
|| As we know,

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

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|| 01:37
|| '''''a''''' is the '''real''' part; '''''b i''''' is imaginary part<nowiki>; </nowiki>'''a''' and '''b''' are constants.

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|| 01:44
|| '''''i''''' is imaginary number and is equal to square root of minus 1.

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|| 01:50
|| In the XY plane, '''a plus b i''' corresponds to the point '''a comma b'''.

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|| 01:57
|| In the '''complex plane''', x axis is called real axis, y axis is called imaginary axis.
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|| 02:05
|| '''Complex numbers, complex plane'''

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|| 02:08
|| In '''complex plane''', '''''z''''' is a '''vector'''.

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|| 02:12
|| Its real axis coordinate is ‘'''a'''’ and imaginary axis coordinate is '''b'''.

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|| 02:19
|| The length of the '''vector 'z'''' is equal to the '''absolute value''' of '''''z''''' and to '''''r'''''.

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|| 02:26
|| According to '''Pythagoras’ theorem''', ''r'' is equal to square root of a squared plus b squared.
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|| 02:35
|| I have already opened '''GeoGebra''' interface.
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|| 02:40
|| Click on ''' Slider''' tool and then in '''Graphics view'''.
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|| 02:46
|| '''Slider''' dialog-box appears.
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|| 02:49
|| By default, '''Number''' radio-button is selected.
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|| 02:53
|| In the '''Name '''field, type '''a'''.
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|| 02:57
|| Set '''Min''' value as 1, '''Max ''' value as 5 and Increment as 1.
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|| 03:07
|| Click '''OK''' button.
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|| 03:10
|| This creates a number '''slider''' named '''a'''.
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|| 03:14
|| Using the '''slider''', '''a''' can have values from 1 to 5, in increments of 1.
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|| 03:23
|| Following the same steps, create '''sliders''' '''b''' and '''c'''.
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|| 03:29
|| In '''input bar''', type the following line.

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|| 03:33
|| '''f x''' in parentheses '''colon equals a space x caret 2 plus b space x plus c'''.

Press '''Enter'''.

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|| 03:48
|| Drag boundary to see '''Algebra''' view properly.

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|| 03:53
|| Pay attention to the spaces indicating multiplication.
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|| 03:59
|| Observe the equation for '''f of x''' in '''Algebra view'''.
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|| 04:04
|| Set '''slider a''' at '''1''', '''slider b''' at minus '''2''' and '''slider''' '''c ''' at minus ''' 3'''.
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||04:17
|| The equation f of x equals 1 x squared minus 2 x minus 3 appears in '''Algebra view'''.
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||04:26
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool.

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|| 04:32
|| Click in '''Graphics''' view.
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|| 04:35
|| Click on '''Move Graphics View''' tool and drag '''Graphics''' view to see parabola '''f'''.
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|| 04:45
|| Function '''f''' is a parabola, intersecting x axis at minus 1 comma 0 and 3 comma 0.

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|| 04:55
|| Thus, '''roots''' of '''fx''' equals x squared minus 2x minus 3 are x equals minus 1 and '''3'''.
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|| 05:05
|| In '''input bar''', type '''Root f''' in parentheses and press '''Enter'''.
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|| 05:13
|| The '''roots''' appear in '''Algebra view'''.

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|| 05:15
|| They also appear as '''x-intercepts''' of the parabola in '''Graphics view'''.
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|| 05:21
|| In '''input bar''', type '''Extremum f''' in parentheses and press '''Enter'''.
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|| 05:30
|| The '''minimum vertex''' appears in '''Algebra''' and '''Graphics views'''.
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|| 05:37
|| After double clicking on point '''C''' in '''Graphics View''', select '''Object Properties'''.
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|| 05:45
|| From '''Color''' tab, change the color to red.
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|| 05:49
|| Close the '''Preferences '''dialog-box.
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|| 05:53
|| Point '''C''' ('''extremum''' of '''f of x''') is red in '''Algebra''' and '''Graphics views'''.
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|| 06:01
|| Click on '''Move''' tool, set '''slider a''' at '''1''', '''slider b''' at '''5''', '''slider c''' at '''10'''.
|-
|| 06:16
|| The equation '''f of x''' equals 1 x squared plus 5x plus 10 appears in '''Algebra view'''.
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|| 06:25
|| Click in and drag '''Graphics''' view to see this parabola.
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|| 06:31
|| It does not intersect the '''x-axis'''.
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|| 06:34
||Points '''A''' and '''B''' are undefined as the function does not intersect the x axis.
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|| 06:41
|| '''Extremum''' point '''C''' is shown in red in '''Algebra''' and '''Graphics views'''.
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|| 06:48
|| Function '''f of x''' equals x squared plus 5x plus 10 has no '''real roots'''.

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|| 06:56
|| Let us see the '''complex roots''' of this equation.

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|| 07:00
|| Click on '''View''', then on '''Spreadsheet'''.

|-
|| 07:05
|| This opens a spreadsheet on the right side of the '''Graphics view'''.
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|| 07:10
|| Click to close '''Algebra view'''.
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|| 07:14
|| Drag the boundary to see '''Spreadsheet''' view properly.
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|| 07:19
|| Type the following '''labels''' and formulae in the '''spreadsheet'''.
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|| 07:24
|| In '''cell A1''', type within quotes '''b caret 2 minus 4ac''' and press '''Enter.'''

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|| 07:38
|| Drag column to adjust width.

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|| 07:42
|| b squared minus 4ac is also called the '''discriminant'''.

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|| 07:47
|| In cells '''A4''' and '''A5''', type '''Root1''' and '''Root2 ''' and press '''Enter'''.

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|| 07:58
|| In cells '''A9''' and '''A10''', type '''Complex root1''' and '''Complex root2'''.

Press '''Enter'''.
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|| 08:11
|| Drag column to adjust width.
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||08:15
|| In '''cell B1''', type '''b caret 2 minus 4 space a space c''' and press '''Enter'''.
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||08:27
|| The value minus 15 appears in '''cell B1''' corresponding to b squared minus 4 a c for '''f x'''.

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|| 08:36
|| Note: '''Discriminant''' is always negative for quadratic '''functions''' without '''real roots'''.

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|| 08:43
|| In '''cell B3''', type within quotes '''minus b divided by 2a'''.

Press '''Enter.'''
|-
||08:55
|| In '''cell B4''', type '''minus b divided by 2 space a'''.

Press '''Enter'''.

|-
|| 09:04
|| Note the value '''-2.5''' appear in cell '''B4'''.
|-
|| 09:10
|| In '''cell B5''', type '''B4''' and press '''Enter. '''

|-
|| 09:17
|| The value '''-2.5''' appears in cell '''B5 '''also.

|-
|| 09:22
|| In '''cell C3''', type the following line and press '''Enter'''.

|-
|| 09:28
|| Within quotes, '''plus minus sqrt D divided by 2a'''
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|| 09:38
|| In '''cell C4''', type '''sqrt B1''' in parentheses''' divided by 2 space a''' and press '''Enter'''.

|-
|| 09:53
|| Note that a question mark appears in '''cell C4'''.
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|| 09:57
|| In '''cell C5''', type '''minus C4''' and press '''Enter'''.

|-
|| 10:05
|| Again, a question mark appears in '''cell C5'''.

|-
|| 10:09
|| There are no '''real''' solutions to the negative square root of the discriminant'''.

|-
|| 10:14
|| In '''input bar''', type '''b4 plus c4 comma 0''' in parentheses and press '''Enter.'''

|-
|| 10:26
|| This should '''plot''' the '''root''' corresponding to ratio of minus b plus square root of discriminant to 2a.
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|| 10:35
|| In input bar, type '''b5 plus c5 comma 0''' in parentheses and press '''Enter'''.

|-
|| 10:46
|| This should plot the '''root''' corresponding to ratio of minus b minus square root of discriminant to 2a.
|-
|| 10:54
|| '''f x''' equals x squared plus 5x plus 10 has no '''real roots'''.

|-
|| 11:00
|| Hence, the points do not appear in '''Graphics view'''.
|-
|| 11:04
|| Click in and drag ''' Graphics view ''' to see this properly.
|-
|| 11:09
|| In '''cell B9''', type '''minus b divided by 2 space a ''' and press '''Enter'''.
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|| 11:21
|| In '''cell B10''', type ''' B9 ''' and press ''' Enter'''.
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|| 11:27
|| '''Discriminant''' is less than 0 for '''f x''' equals x squared plus 5x plus 10.

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|| 11:33
|| So the opposite sign will be taken to allow calculation of '''roots'''.
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|| 11:39
|| In '''cell C9''', type '''sqrt minus B1''' in parentheses '''divided by 2 space a''' and press '''Enter'''.

1.94 appears in '''C9'''.
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|| 11:57
|| In '''cell C10''', type '''minus C9''' and press '''Enter'''.

'''Minus 1.94''' appears in '''C10'''.
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|| 12:08
|| Click in and drag '''Graphics''' view to see the following '''complex''' '''roots'''.
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|| 12:15
|| In '''input bar''', type '''b9 comma c9''' in parentheses and press '''Enter'''.

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|| 12:25
|| This '''complex root''' has real axis coordinate, minus b divided by 2a.

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|| 12:31
|| Imaginary axis co-ordinate is square root of negative '''discriminant''' divided by 2a.

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|| 12:38
|| In input bar, type '''b10 comma c10''' in parentheses and press '''Enter'''.

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|| 12:48
|| This complex root has '''real axis coordinate, minus b divided by 2a'''.

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|| 12:54
|| Imaginary axis co-ordinate is minus square root of negative '''discriminant''' divided by 2a.
|-
|| 13:01
|| Drag boundary to see '''sliders''' in '''Graphics''' view properly.
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|| 13:07
|| Drag the '''slider b''' to minus 2 and '''slider c''' to minus 3.
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||13:16
|| Click in and drag '''Graphics''' view to see the parabola.
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|| 13:12
|| Note how the parabola changes to the one seen for '''f x''' equals x squared minus 2x minus 3.
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|| 13:29
|| The '''real roots''' plotted earlier for '''f x''' equals x squared minus 2x minus 3 appear now.
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|| 13:36
|| Drag boundary to see '''Spreadsheet''' view.
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|| 13:40
|| As '''roots''' are '''real''', calculations for '''complex roots''' become invalid.
|-
|| 13:47
|| Let us summarize.
|-
|| 13:49
|| In this tutorial, we have learnt to:

Visualize quadratic '''polynomials''', their '''roots''' and '''extrema'''

|-
|| 13:57
|| Use a '''spreadsheet''' to calculate '''roots''' ('''real''' and '''complex''') for quadratic '''polynomials'''
|-
|| 14:04
|| As an assignment:

Drag '''sliders''' to graph different quadratic '''polynomials'''.

Calculate '''roots''' of the '''polynomials'''.
|-
|| 14:13
|| The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
|| 14:21
|| The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
|| 14:30
|| Please post your timed queries on this forum.
|-
|| 14:34
|| '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
|| 14:47
|| This is Vidhya Iyer from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

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

PoojaMoolya: Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- || 00:01 || Welcome to this tutorial on '''Complex Roots of Quadratic Equations'''. |- || 00:07 || In this tutorial, we will lea..."