CAT 2024 Slot 2 · Quant · Question 13

Name the two square roots: let u=\sqrt{x+6\sqrt{2}} and v=\sqrt{x-6\sqrt{2}}, so u-v=2\sqrt{2}.

Squaring each separately, {u}^{2}=x+6\sqrt{2} and {v}^{2}=x-6\sqrt{2}.

Subtracting, {u}^{2}-{v}^{2}=12\sqrt{2}.

But {u}^{2}-{v}^{2}=\left( u-v \right)\left( u+v \right), and u-v is already known.

So 2\sqrt{2}\,\left( u+v \right)=12\sqrt{2}, giving u+v=6.

Solving u-v=2\sqrt{2} and u+v=6 together, u=3+\sqrt{2} and v=3-\sqrt{2}.

Then {u}^{2}={\left( 3+\sqrt{2}\right)}^{2}=9+6\sqrt{2}+2=11+6\sqrt{2}.

Matching with {u}^{2}=x+6\sqrt{2} gives x=11.

A quick check: {v}^{2}={\left( 3-\sqrt{2}\right)}^{2}=11-6\sqrt{2}, exactly as required.

Shortcut. Multiplying the given difference by the conjugate sum turns the surds into the plain number 12\sqrt{2}, so the sum of the two roots comes out instantly. Adding the two equations and squaring is a one-line finish.