Question

The equation whose roots are the square of the roots of $x^2+4x+5=0$ is

• A. $x^2-6x+25=0$
• B. $x^2+10x+25=0$
• C. $x^2-26x+25=0$
• D. $x^2+6x+25=0$

Answer 1

The correct option is A $x^2-6x+25=0$

Let $\alpha ,\beta$ be the roots of $f\left(x\right)=x^2+4x+5=0$
Thus, equation whose root are ${\alpha }^2,{\beta }^2$ is,
$f\left(\sqrt{x}\right)=0\Rightarrow (\sqrt{x}{)}^2+4\sqrt{x}+5=0$
$\Rightarrow x+5=-4\sqrt{x}$
Squaring on both sides
$\Rightarrow (x+5{)}^2=(4\sqrt{x}{)}^2$
$\Rightarrow x^2+10x+25=16x$
$\Rightarrow x^2-6x+25=0$