Applications-of-GeoGebra/C2/Complex-Roots-of-Quadratic-Equations/English-timed
From Script | Spoken-Tutorial
Time | Narration |
00:01 | Welcome to this tutorial on Complex Roots of Quadratic Equations. |
00:07 | In this tutorial, we will learn to |
00:10 | Plot graphs of quadratic functions |
00:14 | Calculate real and complex roots of quadratic functions. |
00:19 | To follow this tutorial, you should be familiar with: |
00:22 | GeoGebra interface |
00:25 | Basics of quadratic equations, geometry and graphs |
00:30 | Previous tutorials in this series |
00:33 | If not, for relevant tutorials, please visit our website. |
00:38 | Here I am using:
Ubuntu Linux OS version 14.04, Geogebra 5.0.388.0 hyphen d |
00:53 | Quadratic polynomials |
00:56 | Let us find out more about a 2^{nd} degree polynomial. |
01:01 | y equals a x squared plus b x plus c |
01:06 | The function graphs as a parabola. |
01:09 | If the parabola intersects the x axis, the intercepts are real roots. |
01:15 | If the parabola does not intersect x axis at all, it has no real roots.
Roots are complex. |
01:24 | Let us look at complex numbers. |
01:27 | Complex numbers, XY plane |
01:30 | As we know,
A complex number is expressed as z equals a plus b i. |
01:37 | a is the real part; b i is imaginary part; a and b are constants. |
01:44 | i is imaginary number and is equal to square root of minus 1. |
01:50 | In the XY plane, a plus b i corresponds to the point a comma b. |
01:57 | In the complex plane, x axis is called real axis, y axis is called imaginary axis. |
02:05 | Complex numbers, complex plane |
02:08 | In complex plane, z is a vector. |
02:12 | Its real axis coordinate is ‘a’ and imaginary axis coordinate is b. |
02:19 | The length of the vector 'z' is equal to the absolute value of z and to r. |
02:26 | According to Pythagoras’ theorem, r is equal to square root of a squared plus b squared. |
02:35 | I have already opened GeoGebra interface. |
02:40 | Click on Slider tool and then in Graphics view. |
02:46 | Slider dialog-box appears. |
02:49 | By default, Number radio-button is selected. |
02:53 | In the Name field, type a. |
02:57 | Set Min value as 1, Max value as 5 and Increment as 1. |
03:07 | Click OK button. |
03:10 | This creates a number slider named a. |
03:14 | Using the slider, a can have values from 1 to 5, in increments of 1. |
03:23 | Following the same steps, create sliders b and c. |
03:29 | In input bar, type the following line. |
03:33 | f x in parentheses colon equals a space x caret 2 plus b space x plus c.
Press Enter. |
03:48 | Drag boundary to see Algebra view properly. |
03:53 | Pay attention to the spaces indicating multiplication. |
03:59 | Observe the equation for f of x in Algebra view. |
04:04 | Set slider a at 1, slider b at minus 2 and slider c at minus 3. |
04:17 | The equation f of x equals 1 x squared minus 2 x minus 3 appears in Algebra view. |
04:26 | Under Move Graphics View, click on Zoom Out tool. |
04:32 | Click in Graphics view. |
04:35 | Click on Move Graphics View tool and drag Graphics view to see parabola f. |
04:45 | Function f is a parabola, intersecting x axis at minus 1 comma 0 and 3 comma 0. |
04:55 | Thus, roots of fx equals x squared minus 2x minus 3 are x equals minus 1 and 3. |
05:05 | In input bar, type Root f in parentheses and press Enter. |
05:13 | The roots appear in Algebra view. |
05:15 | They also appear as x-intercepts of the parabola in Graphics view. |
05:21 | In input bar, type Extremum f in parentheses and press Enter. |
05:30 | The minimum vertex appears in Algebra and Graphics views. |
05:37 | After double clicking on point C in Graphics View, select Object Properties. |
05:45 | From Color tab, change the color to red. |
05:49 | Close the Preferences dialog-box. |
05:53 | Point C (extremum of f of x) is red in Algebra and Graphics views. |
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. |
06:25 | Click in and drag Graphics view to see this parabola. |
06:31 | It does not intersect the x-axis. |
06:34 | Points A and B are undefined as the function does not intersect the x axis. |
06:41 | Extremum point C is shown in red in Algebra and Graphics views. |
06:48 | Function f of x equals x squared plus 5x plus 10 has no real roots. |
06:56 | Let us see the complex roots of this equation. |
07:00 | Click on View, then on Spreadsheet. |
07:05 | This opens a spreadsheet on the right side of the Graphics view. |
07:10 | Click to close Algebra view. |
07:14 | Drag the boundary to see Spreadsheet view properly. |
07:19 | Type the following labels and formulae in the spreadsheet. |
07:24 | In cell A1, type within quotes b caret 2 minus 4ac and press Enter. |
07:38 | Drag column to adjust width. |
07:42 | b squared minus 4ac is also called the discriminant. |
07:47 | In cells A4 and A5, type Root1 and Root2 and press Enter. |
07:58 | In cells A9 and A10, type Complex root1 and Complex root2.
Press Enter. |
08:11 | Drag column to adjust width. |
08:15 | In cell B1, type b caret 2 minus 4 space a space c and press Enter. |
08:27 | The value minus 15 appears in cell B1 corresponding to b squared minus 4 a c for f x. |
08:36 | Note: Discriminant is always negative for quadratic functions without real roots. |
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 |
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. |
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. |
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. |
11:21 | In cell B10, type B9 and press Enter. |
11:27 | Discriminant is less than 0 for f x equals x squared plus 5x plus 10. |
11:33 | So the opposite sign will be taken to allow calculation of roots. |
11:39 | In cell C9, type sqrt minus B1 in parentheses divided by 2 space a and press Enter.
1.94 appears in C9. |
11:57 | In cell C10, type minus C9 and press Enter.
Minus 1.94 appears in C10. |
12:08 | Click in and drag Graphics view to see the following complex roots. |
12:15 | In input bar, type b9 comma c9 in parentheses and press Enter. |
12:25 | This complex root has real axis coordinate, minus b divided by 2a. |
12:31 | Imaginary axis co-ordinate is square root of negative discriminant divided by 2a. |
12:38 | In input bar, type b10 comma c10 in parentheses and press Enter. |
12:48 | This complex root has real axis coordinate, minus b divided by 2a. |
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. |
13:07 | Drag the slider b to minus 2 and slider c to minus 3. |
13:16 | Click in and drag Graphics view to see the parabola. |
13:12 | Note how the parabola changes to the one seen for f x equals x squared minus 2x minus 3. |
13:29 | The real roots plotted earlier for f x equals x squared minus 2x minus 3 appear now. |
13:36 | Drag boundary to see Spreadsheet view. |
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. |
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14:47 | This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. |