# The number of real solutions of

Question:

The number of real solutions of $\left|2 x-x^{2}-3\right|=1$ is

(a) 0

(b) 2

(c) 3

(d) 4

Solution:

(b) 2

Given equation: $\left|2 x-x^{2}-3\right|=1$

(i) $2 x-x^{2}-3=1$

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

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

$\Rightarrow(x-2)^{2}=0$

$\Rightarrow x=2,2$

(ii) $-2 x+x^{2}+3=1$

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

$\Rightarrow x^{2}-2 x+1+1=0$

$\Rightarrow(x-1)^{2}-i^{2}=0$

$\Rightarrow(x-1+i)(x-1-i)=0$

$\Rightarrow x=1-i, 1+i$

Hence, the real solutions are 2,2 .