GATE CS 2021 Set 2 Q.4 (GA Q.4): Quantitative Aptitude

If $\left(x - \frac{1}{2}\right)^2 - \left(x - \frac{3}{2}\right)^2 = x + 2$ , then the value of $x$ is:

Official answer and solution

Solution

Expand the difference of two squares to solve the linear equation.

We are given the equation:

\left(x - \frac{1}{2}\right)^2 - \left(x - \frac{3}{2}\right)^2 = x + 2

Apply the identity $A^2 - B^2 = (A - B)(A + B)$ where $A = x - 1/2$ and $B = x - 3/2$ :

A - B = \left(x - \frac{1}{2}\right) - \left(x - \frac{3}{2}\right) = 1

A + B = \left(x - \frac{1}{2}\right) + \left(x - \frac{3}{2}\right) = 2x - 2

Thus the left-hand side is $1 \cdot (2x - 2) = 2x - 2$ . Setting this equal to the right-hand side:

2x - 2 = x + 2 \implies x = 4

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Paper: GATE 2021 Computer Science and Information Technology - Session 2 (CS2)
Question: 4 of 65, printed as GA Q.4
Section: General Aptitude

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.