|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:|
|00:25||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
|00:56||Let us find out more about a 2nd 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.
|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.
|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.
|08:55|| In cell B4, type minus b divided by 2 space a.
|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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