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

The question is about finding Possible real Solutions of a Quadratic Equation. We need to find out the possible real number solutions that satisy the given quadratic equation involving a modulus. Framing and solving equations is an integral part of Linear Equations and Quadratic Equations. Get as much practice as you can in these two topics because the benefits of being good at framing equations will be useful in other concepts.

Question 12: How many real solutions are there for the equation x² – 7|x| - 30 = 0?

1. 3
2. 1
3. 2
4. none

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Quadratic Equations: How many solutions are there for this quadratic equation containing modulus?

Case 1: x > 0
x² – 7x - 30 = 0
( x - 10 ) ( x - 3 ) = 0
X = 10, x = -3
But x = -3 is not possible as we have considered x > 0, thus 1 solution for this case.

Case 2: x < 0
x² + 7x - 30 = 0
( x + 10 )( x - 3)=0
X = -10 and x = 3
Only x = -10 is permissible.
Thus this equation has 2 real solutions

Alternatively, we can think of the above as a quadratic in |x|
x² – 7|x| - 30 = 0 can be factorized as
(|x| -10) ( |x| + 3) = 0
|x| cannot be -3, |x| can only be 10. X can take 2 real values.

The question is "How many real solutions are there for the equation x² – 7|x| - 30 = 0?"

Hence the answer is "2"

Choice C is the correct answer.