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Radical Equation Simplification

CAT 2023 Slot 1 · Quantitative Ability · Medium · Algebra

This is a medium Quantitative Ability question from the CAT 2023 Slot 1 paper. It tests Algebra. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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If 5x+9+5x−9=3(2+2)\sqrt{5x + 9} + \sqrt{5x - 9} = 3(2 + \sqrt{2}), then 10x+9\sqrt{10x + 9} is equal to

  1. A.

    454\sqrt{5}

  2. B.

    272\sqrt{7}

  3. C.

    3313\sqrt{31}

  4. D.

    373\sqrt{7}

Answer

D

Explanation

We have: 5x+9+5x−9=6+32=36+18\sqrt{5x + 9} + \sqrt{5x - 9} = 6 + 3\sqrt{2} = \sqrt{36} + \sqrt{18} By inspection, matching terms: 5x+9=36  ⟹  5x+9=36  ⟹  5x=27\sqrt{5x + 9} = \sqrt{36} \implies 5x + 9 = 36 \implies 5x = 27 Check second term: 5x−9=27−9=18=32\sqrt{5x - 9} = \sqrt{27 - 9} = \sqrt{18} = 3\sqrt{2}, which matches.

Now, calculate 10x+9\sqrt{10x + 9}: 10x+9=2(5x)+9=2(27)+9=63=37\sqrt{10x + 9} = \sqrt{2(5x) + 9} = \sqrt{2(27) + 9} = \sqrt{63} = 3\sqrt{7}

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