Mathematico | Mathematico

The roots of

\frac{1}{(x + 5)^{2}} + \frac{2}{x + 5} + 3 = 4

can be expressed in the form $a \pm \sqrt{b}$ for some $a, b \in Z$ .

Find the value of $a b$ .

Hint

Rearranging feels a bit grim, but it simplifies down very nicely.

Hint

Multiply both sides by $(x + 5)^{2}$ , then expand and simplify.

Solution

\begin{aligned} \frac{1}{(x + 5)^{2}} + \frac{2}{x + 5} + 3 & = 4 \\ (x + 5)^{2} - 2 (x + 5) - 1 & = 0 \\ x^{2} + 10 x + 25 - 2 x - 10 - 1 & = 0 \\ x^{2} + 8 x + 14 & = 0 \\ (x + 4)^{2} - 16 + 14 & = 0 \\ x + 4 & = \pm \sqrt{2} \\ x & = - 4 \pm \sqrt{2} \end{aligned}