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FORMATION OF THE QUADRATIC EQUATION WHOSE ROOTS ARE GIVEN:

Let a quadratic equation be ax2 + bx + c = 0 [a ≠ 0]. Let the roots of this equation be Alpha and ß.


Hence Alpha + ß = and Alphaß = c/a.

Now, ax2 + bx + c = 0 or x2 + -b/ax + c/a = 0 [Since a ≠ 0]

Or x2 – (Alpha + ß)x + Alpha ß =0[Since Alpha + ß = -b/aand Alphaß = c/a]

Or x2 – (sum of the roots)x + product of the roots = 0            (1)

Equation (1) is known as the formula (1) which is used for the formation of a quadratic equation when its roots are given.


For example, if we have to form a quadratic equation whose roots are 2 and -3. Then by formula (1) we get the required equation as given below:


x2 – [2 + (-3)]x + 2.(-3) = 0 or x2 – (-1)x + (-6) = 0 or x2 + x – 6 = 0.

Example 1:

If Alpha and ß be the roots of the equation x2 + 3x + 4 = 0, then find the equation whose roots are (Alpha + ß)2 and (Alpha – ß)2.

Solution:

x2 + 3x + 4 = 0            (1)

Given that the roots of equation (1) areAlpha and ß. Hence Alpha + ß = -3 and Alphaß = 4.

Now, the sum of the roots of the required equation = (Alpha + ß)2 + (Alpha – ß)2

                                                                                   = (-3)2 + [(Alpha + ß)2 - 4 Alphaß]

                                                                                   = 9 + [(-3)2 – 4.4]

                                                                                   = 9 + 9 – 16 = 2

And the product of the roots of the required equation = (Alpha + ß) 2.(Alpha – ß)2

                                                                                       = (-3)2. [(-3)2 – 4.4]

                                                                                       = 9.(-7) = -63

Hence, the required equation is,

x2 – (sum of the roots)x + product of the roots = 0

or x2 – 2x + (-63) = 0

or x2 – 2x - 63 = 0            (Answer)

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