If x = {2}+{3}, find x³-3x.

Correct Answer is : 82+638{2}+6{3}82+63

Correct Answer is : 82+638{2}+6{3}82+63

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SSC CHSL 15 Nov 2025 Shift 1 Paper

Question : 71 of 100

Marks: +1, -0

If x =

\sqrt{2}+\sqrt{3}

x^{3}-3x

Solution:

Concept:

Use the identity

(a+b)^3 = a^3 + b^3 + 3ab(a+b)

to expand

x^3

, then subtract

3x

Explanation:

Given

x = \sqrt{2} + \sqrt{3}

First, find

x^3

using the identity with

a=\sqrt{2}

b=\sqrt{3}

x^3 = (\sqrt{2})^3 + (\sqrt{3})^3 + 3(\sqrt{2})(\sqrt{3})(\sqrt{2}+\sqrt{3})

x^3 = 2\sqrt{2} + 3\sqrt{3} + 3\sqrt{6}(\sqrt{2}+\sqrt{3})

x^3 = 2\sqrt{2} + 3\sqrt{3} + 3\sqrt{12} + 3\sqrt{18}

Simplify

\sqrt{12}=2\sqrt{3}

\sqrt{18}=3\sqrt{2}

x^3 = 2\sqrt{2} + 3\sqrt{3} + 6\sqrt{3} + 9\sqrt{2}

x^3 = 11\sqrt{2} + 9\sqrt{3}

Now,

3x = 3\sqrt{2} + 3\sqrt{3}

Then

x^3 - 3x = (11\sqrt{2} + 9\sqrt{3}) - (3\sqrt{2} + 3\sqrt{3})

(11\sqrt{2} - 3\sqrt{2}) + (9\sqrt{3} - 3\sqrt{3})

8\sqrt{2} + 6\sqrt{3}

Answer:

8\sqrt{2}+6\sqrt{3}

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