The sum of all the roots of the equation |x^2-8x+15|-2x+7=0 is :

Question

The sum of all the roots of the equation $|{x}^{2}-8x+15|-2x+7=0$ is :

Accepted Answer

$\begin{gathered} |{x}^{2}-8x+15|-2x+7=0 \ \Rightarrow |(x-3)(x-5)|-2x+7=0 \end{gathered}$

Now, when $x\le 3$ or $x\ge 5$, then

$\begin{gathered} {x}^{2}-8x+15-2x+7=0 \ \Rightarrow {x}^{2}-10x+22=0 \ \Rightarrow {x}^{2}-10x+25-3=0 \ \Rightarrow (x-5{)}^{2}-3=0 \ \Rightarrow (x-5)=\pm \sqrt{3} \ \Rightarrow x=5+\sqrt{3},5-\sqrt{3} \end{gathered}$

Since, $x\le 3$ or $x\ge 5$

$	herefore x=5+\sqrt{3}$

When, $3<x<5$, then

$\begin{gathered} -({x}^{2}-8x+15)-2x+7=0 \ \Rightarrow -{x}^{2}+8x-15-2x+7=0 \ \Rightarrow -{x}^{2}+6x-8=0 \ \Rightarrow {x}^{2}-6x+8=0 \ \Rightarrow (x-4)(x-2)=0\Rightarrow x=2,4 \end{gathered}$

Since, $3<x<5$

$	herefore x=4$

$	herefore =(5+\sqrt{3})+4=9+\sqrt{3}$

Solutions · 1