Quadratic Equations
Nested Absolute Value Equation — √(β/α)+√(αβ)
nta_pyq_2026_jan
Grade 11
Question:
If $\alpha$ and $\beta$ ($\alpha<\beta$) are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0$, $x\geq0$, then $\sqrt{\dfrac{\beta}{\alpha}}+\sqrt{\alpha\beta}$ is equal to:
Step-by-Step Solution
Key Concept: Let $t=\sqrt{x}\geq0$. Equation: $(-2+\sqrt{3})|t-3|+(t^2-6t)+(9-2\sqrt{3})=0$. Since $t^2-6t+9=(t-3)^2$: $(t-3)^2+(\sqrt{3}-2)|t-3|-2\sqrt{3}=0$. Let $u=|t-3|\geq0$: $u^2+(\sqrt{3}-2)u-2\sqrt{3}=0$.
$\alpha=1$, $\beta=25$. $\sqrt{\beta/\alpha}+\sqrt{\alpha\beta}=10$.
Correct Answer: 4