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

Question

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

Accepted Answer

$$
\begin{aligned}
& \text { We have, }\left|x^2-8 x+15\right|-2 x+7=0 \\
& \Rightarrow |(x-3)(x-5)|-2 x+7=0
\end{aligned}
$$

Now, when $x \leq 3$ or $x \geq 5$, then

$$
\[\begin{gathered} x^2-8 x+15-2 x+7  =0 \ \Rightarrow  x^2-10 x+22=0 \ \Rightarrow  x^2-10 x+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, $x \leq 3$ or $x \geq 5$

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

When, $3 < x < 5$, then

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

Since, $3 < x < 5$

$$
\therefore x=4
$$

$$
\therefore \text { Sum of roots }=(5+\sqrt{3})+4=9+\sqrt{3}
$$

Solutions · 1