Question

The real roots of the equation $|x^2+4x+3|+2x+5=0$ are

• A. $4,-1+\sqrt{3}$
• B. $-4,-1-\sqrt{3}$
• C. $-6,-1$
• D. $6,-1$

Answer 1

The correct option is B $-4,-1-\sqrt{3}$

$|x^2+4x+3|+2x+5=0$
$|(x+1)(x+3)|+2x+5=0$
Now from the expression $(x+1)(x+3)$, we can see that the expression is negative for values lying between $-1$ and $-3$.
So, for values not lying between $-1$ and $-3$ we can write equation as $x^2+6x+8=0$
$(x+2)(x+4)=0$
$x$ cannot be equal to $-2$ as $x$ cannot lie between $-1$ and $-3$
$\therefore x=-4$
For values in the range of $-3$ to $-1$ we can write equation as
$-x^2-4x-3+2x+5=0$
$-x^2-2x+2=0$
$x^2+2x-2=0$
$x=\frac{-2\pm \sqrt{12}}{2}$
$x=\sqrt{3}-1$ is discarded since it lies outside the $-3$ to $-1$
So, $x=-1-\left(\sqrt{3}\right)$ and $-4$