Difference between revisions of "Applications-of-GeoGebra/C2/Complex-Roots-of-Quadratic-Equations/English"
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|| 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 '''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. | 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. | || 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'''. | Hence, the points do not appear in '''Graphics view'''. | ||
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|| 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 | + | 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. | Imaginary axis co-ordinate is square root of negative '''discriminant''' divided by 2a. | ||
Latest revision as of 12:45, 5 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 square root 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 square root 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 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 >> 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 >> 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 Preferences dialog-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 >> 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. |
| Point to f(x) in Algebra view. | 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 of Graphics view. | 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 square root 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 square root of discriminant to 2a. |
| Point to the graph. | f x equals x squared plus 5x plus 10 has no real roots.
Hence, the points do not appear in Graphics view. |
| Click in >> drag Graphics view. | 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 >> press Enter. | In cell B10, type B9 and press Enter. |
| point to the cell. | 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 >> 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 square root of negative discriminant divided by 2a. |
| Type (b10,c1 0) 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 square root 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 >> 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. |
