Question

If roots of the equation $(x-12{)}^2+(2x-24\left)\right(x+12)+(x+12{)}^2=0$ are decreased by $12$, then the roots of the transformed equation are

• A. irrational roots
• B. rational and distinct roots
• C. rational and equal roots
• D. non real roots

Answer 1

The correct option is C rational and equal roots

$f\left(x\right)=(x-12{)}^2+2(x-12\left)\right(x+12)+(x+12{)}^2$
Roots are decreased by $12$, so the transformed equation is,
$f(x+12)=0$
$\Rightarrow x^2+2x(x+24)+(x+24{)}^2=0\Rightarrow x^2+2x^2+48x+x^2+48x+{24}^2=0\Rightarrow 4x^2+96x+{24}^2=0\Rightarrow x^2+24x+{12}^2=0\Rightarrow (x+12{)}^2=0\Rightarrow x=-12,-12$

Hence, roots are rational and equal.