Question

If the equation $x^2+x-12=0$ has roots as $\alpha ,\beta$, then the equation with the roots ${\alpha }^2,{\beta }^2$ is:

• A. $x^2+25x+144=0$
  
• B. $x^2+25x-144=0$
  
• C. $x^2-25x-144=0$
  
• D. $x^2-25x+144=0$
  

Answer 1

The correct option is D $x^2-25x+144=0$


Given that $x^2+x-12=0$ has roots as $\alpha ,\beta$
$\Rightarrow \alpha +\beta =-1$
$\Rightarrow \alpha .\beta =-12$
Equation with the roots ${\alpha }^2,{\beta }^2$ can be written as,
$x^2-({\alpha }^2+{\beta }^2)x+{\alpha }^2.{\beta }^2=0$

So,
$(\alpha +\beta {)}^2={\alpha }^2+{\beta }^2+2\alpha .\beta =(-1{)}^2=1$
$\Rightarrow {\alpha }^2+{\beta }^2=1-2\alpha .\beta$
$\Rightarrow {\alpha }^2+{\beta }^2=1-2(-12)=1+24=25$
Also
${\alpha }^2{\beta }^2=(\alpha \beta {)}^2=(-12{)}^2=144$

Therefore we can say that the required equation is,
$x^2-\left(25\right)x+144=0$