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Formation Of Quadratic Equation With Given Roots

Formation of Quadratic Equation whose roots are given  Online Tutoring / Homework Help
Let a quadratic equation be ax2 + bx + c = 0 [a ≠ 0]. Let the roots of this equation be Alpha and .


Hence Alpha + = and Alpha = c/a.

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

Or x2 (Alpha + )x + Alpha =0[Since Alpha + = -b/aand Alpha = 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 be the roots of the equation x2 + 3x + 4 = 0, then find the equation whose roots are (Alpha + )2 and (Alpha )2.

Solution:

x2 + 3x + 4 = 0            (1)

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

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

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

                                                                                   = 9 + [(-3)2 4.4]

                                                                                   = 9 + 9 16 = 2

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