Question

The sum of the real solutions of equation \(2|x|^2 + 51 = |1 + 20x|\) is

• A. 5
• B. 24
• C. 0
• D. None of these

Hint:

Always convert \(|x|^2 = x^2\) first, then split modulus carefully into cases.

Answer 1

The Correct Option is D

Concept: \[ |x|^2 = x^2 \Rightarrow 2x^2 + 51 = |1 + 20x| \]
Step 1: Split modulus.
Case 1: \(1 + 20x \ge 0\) \[ 2x^2 + 51 = 1 + 20x \] \[ 2x^2 - 20x + 50 = 0 \Rightarrow x^2 - 10x + 25 = 0 \Rightarrow (x - 5)^2 = 0 \Rightarrow x = 5 \] Check condition: \[ 1 + 20(5) = 101 > 0 \Rightarrow \text{valid} \]
Case 2: \(1 + 20x < 0\) \[ 2x^2 + 51 = -(1 + 20x) \] \[ 2x^2 + 20x + 52 = 0 \Rightarrow x^2 + 10x + 26 = 0 \] \[ D = 100 - 104 = -4 < 0 \Rightarrow \text{no real roots} \]
Step 2: Sum of real solutions. \[ \text{Only solution } x = 5 \]
Step 3: Final check. \[ \text{Sum} = 5 \] It is a mismatch with given options. Therefore, the correct answer is (D) None of these.