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