There, We presented the Derivation Of Formulae Relating roots and coefficients of a Quadratic Equation. Solved Examples on applying the formulae are given. Problems for practice are also given in Exercise.

That knowledge is a prerequisite here.

Here we deal with finding the Quadratic Equation whose roots are given.

We explain the Formula and apply it in solving problems. Exercise problems are also given for practice.

To find the Quadratic Equation whose roots are given :

The Quadratic Equation whose roots are given is x^{2} - (sum of the roots)x + (product of the roots) = 0

PROOF :

We know

Sum of the roots = α + β = -b⁄a = -{(coefficient of x)⁄(coefficient of x^{2})}

Product of the roots = αβ = c⁄a = (constant term)⁄(coefficient of x^{2})}

Substituting these in the LHS of the Equation above and multiplying through out with a , we get the equation as ax^{2} + bx + c = 0 which is the original Equation. (Proved.)

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Find the quadratic equation whose roots are 3, -2.

Solution of Example 1 of Quadratic Equation Solver :

The given roots are 3, -2 Sum of the roots = 3 + (-2) = 3 - 2 = 1; Product of the roots = 3 x (-2) = -6 We know the Quadratic Equation whose roots are given is x^{2} - (sum of the roots)x + (product of the roots) = 0 ∴ The required equation is x^{2} - (1)x + (-6) = 0 i.e x^{2} - x - 6 = 0 Ans.

Example 2 : Quadratic Equation Solver

Find the quadratic equation whose roots are 4⁄3, 1⁄3.

Solution of Example 2 of Quadratic Equation Solver :

The given roots are 4⁄3, 1⁄3 Sum of the roots = 4⁄3 + 1⁄3 = 5⁄3; Product of the roots = 4⁄3 x 1⁄3 = 4⁄9 We know the Quadratic Equation whose roots are given is x^{2} - (sum of the roots)x + (product of the roots) = 0 ∴ The required equation is x^{2} - (5⁄3)x + (4⁄9) = 0 Multiplying both sides with 9, we get 9x^{2} -15x + 4 = 0 Ans.

Example 3 : Quadratic Equation Solver

Find the quadratic equation whose roots are lm, mn.

Solution of Example 3 of Quadratic Equation Solver :

The given roots are lm, mn Sum of the roots = lm + mn = m(l + n); Product of the roots = lm x mn = lm^{2}n We know the Quadratic Equation whose roots are given is x^{2} - (sum of the roots)x + (product of the roots) = 0 ∴ The required equation is x^{2} - m(l + n)x + lm^{2}n = 0. Ans.

Find the quadratic equation whose roots are 5 + √7), (5 - √7).

Solution of Example 4 of Quadratic Equation Solver :

The given roots are 5 + √7, 5 - √7 Sum of the roots = (5 + √7) + (5 - √7) = 10; Product of the roots = (5 + √7)x(5 - √7) = 5^{2} - (√7)^{2} = 25 - 7 = 18 We know the Quadratic Equation whose roots are given is x^{2} - (sum of the roots)x + (product of the roots) = 0 ∴ The required equation is x^{2} - (10)x + (18) = 0 i.e x^{2} - 10x + 18 = 0 Ans.

Exercise : Quadratic Equation Solver

Find the quadratic equation whose roots are -2, -4.

Find the quadratic equation whose roots are 3 + √2, 3- √2

Find the quadratic equation whose roots are (l - m), (l + m)

Find the quadratic equation whose roots are 1/2, 3/2

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