cat 2023 Question 3

Question

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

Accepted Answer

Given,

$ \sqrt{5x+9} + \sqrt{5x - 9} = 3(2 + \sqrt{2}) $

$ \sqrt{5x+9} + \sqrt{5x-9} = 6 + 3\sqrt{2} $

$ \sqrt{5x+9} + \sqrt{5x-9} = \sqrt{36} + \sqrt{18} $

Comparing the L.H.S. and R.H.S.

$ 5x+9=36 $ => $ 5x=27 $ => $ x=\dfrac{27}{5} $ (can be verified using the second term as well).

$ \sqrt{10x+9} = \sqrt{\left(10 imes\dfrac{27}{5} ight)+9} = \sqrt{63}=3\sqrt{7} $

Question 3.