Quadratic Equations
Absolute Value Equation — Case Analysis
nta_pyq_2023_apr
Grade 11
Question:
The sum of all the roots of $|x^2-8x+15|-2x+7=0$ is
$9-\sqrt{3}$
$9+\sqrt{3}$
$11-\sqrt{3}$
$11+\sqrt{3}$
Step-by-Step Solution
Key Concept: Case I ($x\leq3$ or $x\geq5$): $x^2-10x+22=0\Rightarrow x=5\pm\sqrt{3}$. Only $x=5+\sqrt{3}$ satisfies. Case II ($3<x<5$): $x^2-6x+8=0\Rightarrow x=4$ (valid).
Roots: $5+\sqrt{3}$ and $4$. Sum $=9+\sqrt{3}$.
Correct Answer: 2