CAT Quantitative Aptitude Questions | CAT Algebra - Linear Equations; Quadratic Equations

The question is about Number of Solutions of a Quadratic Equation. We need to find out the real numbers which satisfy the given quadratic equations involving modulus. Solving questions from Linear and Quadratic Equations well is an integral part to cracking CAT. Get cartloads of practice on these two topics.

Question 4: What is the number of real solutions of the equation x² - 7|x| - 18 = 0?

1. 2
2. 4
3. 3
4. 1

Video Explanation

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Quadratic Equations: Modulus of a number cannot be negative. A simple idea that somehow escapes our mind every now and then.

Let us split this into two cases. Case 1, when x is greater than 0 and Case 2, when x is lesser than 0.
Case 1
x < 0. Now, |x| = x
x² – 7x – 18 = 0
(x – 9) (x + 2) = 0
x is either –2 or +9.

Case 2
x < 0. Now, |x| = –x
x² + 7x – 18 = 0
(x + 9) (x – 2) = 0
x is either –9 or +2.

However, in accordance with the initial assumption that x < 0, x can only be –9 (cannot be +2).
Hence, this equation has two roots: –9 and +9.

Alternatively, we can treat this as a quadratic in |x|, the equation can be written as |x|² – 7 |x| – 18 = 0.
Or, (|x| – 9) (|x| + 2) = 0
|x| = 9 or –2. |x| cannot be –2.
S|x| = 9, x = 9 or –9.

The question is "What is the number of real solutions of the equation x² - 7|x| - 18 = 0?"

Hence the answer is "2"

Choice A is the correct answer.