Question

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Answer 1

$\star\:\:\bf\large\underline\blue{Question:-}$

• Simplify the problem.

$\star\:\:\bf\large\underline\blue{Solution:-}$

$\sqrt{a + b + 2x + 2 \sqrt{ab + (a + b)x + {x}^{2} } }$

Now, let

$y = \sqrt{ab + (a + b)x + {x}^{2} }$

Therefore,

$\sqrt{a + b + 2x + 2 \sqrt{ab + (a + b)x + {x}^{2} } }$

$= \sqrt{a + b + 2x + 2 + 2 \sqrt{y} }$

Now, we will factorise y.

$\sqrt{ab + (a + b)x + {x}^{2} }$

$= \sqrt{ab + ax + bx + {x}^{2} }$

$= \sqrt{a(b + x) +x(b+ x)}$

$= \sqrt{(x + a)(x + b)}$

Now,

$= \sqrt{a + b + 2x + 2 + 2 \sqrt{y} }$

$= \small\sqrt{(x + a) + (x + b) + 2 \sqrt{(x + a)(x + b)} }$

$= \sqrt{ {( \sqrt{x + a}) }^{2} + {( \sqrt{(x + b} )}^{2} + 2 \times \sqrt{(x + a} \times \sqrt{(x + b)} }$

$= \sqrt{ {( \sqrt{x + a} + \sqrt{x + b} )}^{2} }$

$= \sqrt{x + a} + \sqrt{x + b}$

$\star\:\:\bf\large\underline\blue{Answer:-}$

• $\sqrt{x + a } + \sqrt{x + b}$