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Basics & Logarithms: Sum Real Roots Equation

JEE Maths question with a full step-by-step solution.

Question
The sum of all real roots of the equation ∣x−2∣2+∣x−2∣−2=0|x-2|^2 + |x-2| - 2 = 0 is
Solution
Answer: 4
The sum of all real roots of the equation ∣x−2∣2+∣x−2∣−2=0|x-2|^2 + |x-2| - 2 = 0 is ___. Step 1: Let u=∣x−2∣≥0u = |x-2| \geq 0. The equation becomes:
u2+u−2=0  ⟹  (u+2)(u−1)=0u^2 + u - 2 = 0 \implies (u+2)(u-1) = 0
 u=−2u = -2 (rejected, since u≥0u \geq 0) or u=1u = 1. Step 2: ∣x−2∣=1⇒x−2=±1⇒x=3|x-2| = 1 \Rightarrow x - 2 = \pm 1 \Rightarrow x = 3 or x=1x = 1. Step 3: Sum of roots =3+1=4= 3 + 1 = 4. Answer: 4
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