Given that P = x2 + 4x - 2, Q = 2x - 1 and Q - p = 2, find x

Analyze the question: We are given P = x² + 4x − 2 and Q = 2x − 1, and the condition Q − P = 2 (the "p" in the question is P). We need to find x.

Concept: Substitute the expressions for P and Q into the equation, simplify to a quadratic equation, then solve it.

Solution:

Q − P = 2

(2x − 1) − (x² + 4x − 2) = 2

2x − 1 − x² − 4x + 2 = 2

−x² − 2x + 1 = 2

−x² − 2x − 1 = 0

x² + 2x + 1 = 0

(x + 1)² = 0

x = −1 (a repeated root)

Check:

• P = (−1)² + 4(−1) − 2 = 1 − 4 − 2 = −5
• Q = 2(−1) − 1 = −3
• Q − P = −3 − (−5) = 2 ✓

The answer is **B.

Tip: When you get a quadratic, rearrange it so one side is zero and look for a perfect square such as x² + 2x + 1 = (x + 1)². A repeated root means there is only one answer. You can also test the options by substituting them back into the equation, which is a quick check in the exam hall.