Solve the following equations by factorisation:

$\sqrt{x + 15} = x + 3$

Given,

$\Rightarrow \sqrt{x + 15} = x + 3 \\[1em] \Rightarrow x + 15 = (x + 3)^2 \text{ (On squaring both sides) } \\[1em] \Rightarrow x + 15 = x^2 + 9 + 6x \\[1em] \Rightarrow x^2 + 6x - x + 9 - 15 = 0 \\[1em] \Rightarrow x^2 + 5x - 6 = 0 \\[1em] \Rightarrow x^2 + 6x - x - 6 = 0 \\[1em] \Rightarrow x(x + 6) - 1(x + 6) = 0 \\[1em] \Rightarrow (x - 1)(x + 6) = 0 \\[1em] \Rightarrow x - 1 = 0 \text{ or } x + 6 = 0 \\[1em] \Rightarrow x = 1 \text{ or } x = -6$

Since we squared the equation, so roots need to be checked

Putting x = -6 in equation

$\Rightarrow \sqrt{-6 + 15} = -6 + 3 \\[1em] \Rightarrow \sqrt{ 9 } = -3 \\[1em]$

L.H.S. = 3 and R.H.S. = -3

Since, L.H.S. ≠ R.H.S. hence, x = -6 is not root of the equation

Putting x = 1 in equation

$\Rightarrow \sqrt{1 + 15} = 1 + 3 \\[1em] \Rightarrow \sqrt{ 16 } = 4 \\[1em]$

L.H.S. = R.H.S. = 4

Hence, root of the equation is 1.