# The number of real roots of the equation

Question:

The number of real roots of the equation $\left(x^{2}+2 x\right)^{2}-(x+1)^{2}-55=0$ is

(a) 2

(b) 1

(c) 4

(d) none of these

Solution:

(a) 2

$\left(x^{2}+2 x\right)^{2}-(x+1)^{2}-55=0$

$\Rightarrow\left(x^{2}+2 x+1-1\right)^{2}-(x+1)^{2}-55=0$

$\Rightarrow\left\{(x+1)^{2}-1\right\}^{2}-(x+1)^{2}-55=0$

$\Rightarrow\left\{(x+1)^{2}\right\}^{2}+1-3(x+1)^{2}-55=0$

$\Rightarrow\left\{(x+1)^{2}\right\}^{2}-3(x+1)^{2}-54=0$

Let $p=(x+1)^{2}$

$\Rightarrow p^{2}-3 p-54=0$

$\Rightarrow p^{2}-9 p+6 p-54=0$

$\Rightarrow(p+6)(p-9)=0$

$\Rightarrow p=9 \quad$ or $\quad p=-6$

Rejecting $p=-6$

$\Rightarrow(x+1)^{2}=9$

$\Rightarrow x^{2}+2 x-8=0$

$\Rightarrow x^{2}+4 x-2 x-8=0$

$\Rightarrow(x+4)(x-2)=0$

$\Rightarrow x=2, \quad x=-4$