Quadratic Equations : Sum & Product of roots

Quadratic equations are equations of the form ax² + bx + c = 0.

A quadratic equation has two roots. These roots are found either by factorizing the quadratic equation or by using the formula

Here is a typical quadratic equation question

If m and n are the roots of the quadratic equation x² – (2 √ 5)x – 2 = 0, the value of m² + n² is:

A. 22
B. 24
C. 32
D. 20
E. 18

Correct Answer is Choice B. 24.

Explanation to GMAT Quadratic Equations Practice Question

Solving such GMAT algebra questions requires knowledge of two concepts:
1. Formula to compute the sum and product of the roots of quadratic equations
2. Conversant with commonly used algebraic identities.

The question states that ‘m’ and ‘n’ are roots of the equation. We have to find the value of m² + n²

m² + n² = (m + n)² – 2mn

(m + n), the sum of the roots of a quadratic equation of the form ax² + bx + c = 0 is

mn, the product of the roots of the equation =

The sum of the roots of the equation x² – (2 √ 5)x – 2 = 0 is (2 √ 5).
Product of the roots of the equation = -2.

Hence, (m + n)² – 2mn = (2 √ 5)² – 2(-2) = 20 + 4 = 24.

A few other useful algebraic identities that you should know

m³ + n³ = (m + n)³ – 3mn(m + n)

m³ – n³ = (m – n)³ + 3mn(m – n)

You should compute the values of m³ + n³  and m³ – n³ for these two quadratic equations
1. x² – 5x + 4 = 0
2. x² – 15x -76 = 0