Question

If $|x-7{|}^2-3|x-7|-10=0$, then value(s) of $x$ can be equal to

• A. $2$
• B. $5$
• C. $9$
• D. $12$

Answer 1

Answer: (A), (D)

Given $|x-7{|}^2-3|x-7|-10=0$
Let $|x-7|=t$
$\Rightarrow t^2-3t-10=0$
Here, the middle term $-3t$ expressed as sum of $-5t$ and $2t$ such that their product $-5t\times 2t=-10t^2$ is equal to product of extreme terms $(-10(t^2)=-10t^2)$
$\Rightarrow t^2-5t+2t-10=0$
$\Rightarrow t(t-5)+2(t-5)=0$
​​​​​​​$\Rightarrow (t-5)(t+2)=0$
$\Rightarrow (t-5)=0$ or $t+2=0$
​​​​​​​​​​​​​​$\Rightarrow t=5\text{or}t=-2$
As $|x-7|\ge 0$, so
$\Rightarrow |x-7|=5$
$\Rightarrow x-7=\pm 5$
$\Rightarrow x-7=-5orx-7=5$
$\Rightarrow x=-5+7orx=5+7$
$\therefore x=2,x=12$
Hence, the correct options are A ($2$) and D $(12$)