Question

If $x^2-13x+36=0$ has $p,q$ as its roots, then which of the following is the equation with the roots $\sqrt{p},\sqrt{q}$ ?

• A. $x^2+5x-6=0$
  
• B. $x^2-5x-6=0$
  
• C. $x^2-5x+6=0$
  
• D. $x^2+5x+6=0$
  

Answer 1

The correct option is C $x^2-5x+6=0$


Given that $x^2-13x+36=0$ has $p,q$ as its roots.
$\Rightarrow p+q=13$
$\Rightarrow pq=36\Rightarrow \sqrt{pq}=6$
An equation with roots $\sqrt{p},\sqrt{q}$ can be written as:
$x^2-(\sqrt{p}+\sqrt{q})x+\sqrt{pq}=0$

So $(\sqrt{p}+\sqrt{q}{)}^2=p+q+2\sqrt{pq}$
$\Rightarrow (\sqrt{p}+\sqrt{q}{)}^2=13+2(6)=13+12=25$
$\Rightarrow \sqrt{p}+\sqrt{q}=\sqrt{25}=5$

Therefore the required equation can be written as:
$x^2-5x+6=0$