Question

The solution of the equation ${\left|x+1\right|}^2-\left|x+2\right|-26=0$ is:

• A. $\frac{-1+\sqrt{\left(109\right)}}{2}$,$\frac{-3-\sqrt{\left(101\right)}}{2}$
• B. $-7,\sqrt{29}$
• C. $\pm \sqrt{29}$
• D. $-7,29$

Answer 1

The correct option is A $\frac{-1+\sqrt{\left(109\right)}}{2}$,$\frac{-3-\sqrt{\left(101\right)}}{2}$

$|x+1{|}^2$ will be always non negative so we can expand it and remove modulus.

So, Equation becomes $x^2+2x-25-|x+2|=0$

For $x>-2$

$x^2+x-27=0$ which gives $x=\frac{-1_{-}^{+}\sqrt{\left(109\right)}}{2}$

But since we assumed $x>-2$, $x=\frac{-1+\sqrt{109}}{2}$

Now for $x<-2$

$x^2+3x-23$ which gives x = $\frac{-3_{-}^{+}\sqrt{101}}{2}$

But since we assumed $x<-2$, x = $\frac{-3-\sqrt{101}}{2}$