Question

Solve the following equations:
$\sqrt{3x^2-7x-30}+\sqrt{2x^2-7x-5}=x+5$.

Answer 1

$\sqrt{3x^2-7x-30}-\sqrt{2x^2-7x-5}=x-5\dots \dots \left(1\right)$

The identity, $(3x^2-7x-30)-(2x^2-7x-5)=x^2-25\text{is true for all}x\dots \dots \left(2\right)$

Dividing (2) by (1), we get

$\sqrt{3x^2-7x-30}+\sqrt{2x^2-7x-5}=x+5\dots \dots \left(3\right)$

Adding (1) and (3) we get

$2\sqrt{3x^2-7x-30}=2x$

Squaring both sides, we get

$3x^2-7x-30=x^2⟹2x^2-7x-30=0⟹(2x+5)(x-6)=0⟹x=6,\frac{-5}{2}$