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CAT 2023 Slot 1 — QA Question 19

Square root of SurdsEasy

If 5x+9 + 5x-9 = 3(2 + √2), then find the value of 10x+9?

Answer & solution

  • 3√7

  • B

    3√5

  • C

    4√3

  • D

    7√3

Solution

Easy

The right-hand side 3(2+2)=6+323(2+\sqrt2)=6+3\sqrt2 is itself a sum of two surds, 36+18\sqrt{36}+\sqrt{18}. Match each term to one of the two square roots on the left — that splits one equation into two simple linear ones, and from 5x5x you get 10x+910x+9 directly.

1

Rewrite the right side as a sum of surds:

3(2+2)=6+32=36+183(2+\sqrt2)=6+3\sqrt2=\sqrt{36}+\sqrt{18}
2

Match term by term. Since 5x+9>5x−9\sqrt{5x+9}>\sqrt{5x-9}, pair the larger surd with the larger root:

5x+9=36,5x−9=18\sqrt{5x+9}=\sqrt{36},\qquad \sqrt{5x-9}=\sqrt{18}
3

Square both to get linear equations (either one fixes xx; both agree):

5x+9=36 ⇒ 5x=275x−9=18 ⇒ 5x=27✓\begin{aligned} &5x+9=36\ \Rightarrow\ 5x=27\\ &5x-9=18\ \Rightarrow\ 5x=27 \quad\checkmark \end{aligned}
4

Build the requested quantity from 5x=275x=27:

10x=54 ⇒ 10x+9=6310x+9=63=9⋅7=37\begin{aligned} &10x=54\ \Rightarrow\ 10x+9=63\\ &\sqrt{10x+9}=\sqrt{63}=\sqrt{9\cdot7}=3\sqrt7 \end{aligned}

10x+9=37\sqrt{10x+9}=\mathbf{3\sqrt7}.

Notice the two given surds add and subtract the same 99. Their squares add to (5x+9)+(5x−9)=10x(5x+9)+(5x-9)=10x, so you only ever need 5x5x — no need to isolate xx on its own.

CAT 2023 Slot 1 QA Q19: If 5 x + 9 + 5 x - 9 = 3(2 + √2), then find the value of 10 x + 9 ? — Solution | TheCATExam