If α and β are the two roots of a quadratic equation, then the formula to construct the quadratic equation is x2 - (α + β)x + αβ = 0 That is, x2 - (sum of roots)x + product of roots = 0 If a quadratic equation is given in standard form, we can find the sum and product of the roots using coefficient of x2, x and constant term. Let us consider the standard form of a quadratic equation, ax2 + bx + c = 0 (a, b and c are real and rational numbers) Let α and β be the two zeros of the above quadratic equation. Then the formula to get sum and product of the roots of a quadratic equation : Note : Irrational roots of a quadratic equation occur in conjugate pairs. That is, if (m + √n) is a root, then (m - √n) is the other root of the same quadratic equation equation. Example 1 : Form the quadratic equation whose roots are 2 and 3. Solution : Sum of the roots is = 2 + 3 = 5 Product of the roots is = 2 x 3 = 6 Formation of quadratic equation : x2 - (sum of the roots)x + product of the roots = 0 x2 - 5x + 6 = 0 Example 2 : Form the quadratic equation whose roots are 1/4 and -1. Solution : Sum of the roots is = 1/4 + (-1) = 1/4 - 1 = 1/4 - 4/4 = (1 - 4)/4 = -3/4 Product of the roots is = (1/4) x (-1) = -1/4 Formation of quadratic equation : x2 - (sum of the roots)x + product of the roots = 0 x2 - (-3/4)x + (-1/4) = 0 x2 + (3/4)x - 1/4 = 0 Multiply each side by 4. 4x2 + 3x - 1 = 0 Example 3 : Form the quadratic equation whose roots are 2/3 and 5/2. Solution
: Sum of the roots is = 2/3 + 5/2 The least common multiplication of the denominators 3 and 2 is 6. Make each denominator as 6 using multiplication. Then, = 4/6 + 15/6 = (4 + 15)/6 = 19/6 Product of the roots is = 2/3 x 5/2 = 5/3 Formation of quadratic equation : x2 - (sum of the roots)x + product of the roots = 0 x2 - (19/6)x + 5/3 = 0 Multiply each side by 6. 6x2 - 19x + 10 = 0 Example 4 : If one root of a quadratic equation (2 + √3), then form the equation given that the roots are irrational. Solution : (2 + √3) is an irrational number. We already know the fact that irrational roots of a quadratic equation will occur in conjugate pairs. That is, if (2 + √3) is one root of a quadratic equation, then (2 - √3) will be the other root of the same equation. So, (2 + √3) and (2 - √3) are the roots of the required quadratic equation. Sum of the roots is = (2 + √3) + (2 - √3) = 4 Product of the roots is = (2 + √3)(2 - √3) = 22 - √32 = 4 - 3 = 1 Formation of quadratic equation : x2 - (sum of the roots)x + product of the roots = 0 x2 - 4x + 1 = 0 Example 5 : If α and β be the roots of x2 + 7x + 12 = 0, find the quadratic equation whose roots are (α + β)2 and (α - β)2 Solution : Given : α and β be the roots of x2 + 7x + 12 = 0. Then, sum of roots = -coefficient of x/coefficient of x2 α + β = -7/1 = -7 product of roots = constant term/coefficient of x2 αβ = 12/1 = 12 Quadratic equation with roots (α + β)2 and (α - β)2 is x2 - [(α + β)2 + (α - β)2]x + (α + β)2(α - β)2 = 0 ----(1) Find the values of (α + β)2 and (α - β)2. (α + β)2 = (-7)2 (α + β)2 = 49 (α - β)2 = (α + β)2 - 4αβ (α - β)2 = (-7)2 - 4(12) (α - β)2 = 49 - 48 (α - β)2 = 1 So, the required quadratic equation is (1)----> x2 - [49 + 1]x + 49 ⋅ 1 = 0 x2 - 50x + 49 = 0 Example 6 : If α and β be the roots of x2 + px + q = 0, find the quadratic equation whose roots are α/β and β/α Solution : Given : α and β be the roots of x2 + px + q = 0. Then, sum of roots = -coefficient of x/coefficient of x2 α + β = -p/1 α + β = -p product of roots = constant term/coefficient of x2 αβ = q/1 αβ = q Quadratic equation with roots α/β and β/α is x2 - (α/β + β/α)x + (α/β)(β/α) = 0 x2 - [α/β + β/α]x + 1 = 0 ----(1) Find the value of (α/β + β/α). α/β + β/α = α2/αβ + β2/αβ = (α2 + β2)/αβ = [(α + β)2 - 2αβ]/αβ = (p2 - 2q)/q So, the required quadratic equation is (1)----> x2 -[(p2 - 2q)/q]x + 1 = 0 Multiply each side by q. qx2 - (p2 -
2q)x + q = 0 Kindly mail your feedback to We always appreciate your feedback. ©All rights reserved. onlinemath4all.com How do you write a quadratic equation with the sum and product of roots?For a quadratic equation ax2+bx+c = 0, the sum of its roots = –b/a and the product of its roots = c/a. A quadratic equation may be expressed as a product of two binomials. Here, a and b are called the roots of the given quadratic equation.
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