Question

Which of the following can be the roots of the equation $x^2+px+45=0$ if the squared difference of its roots is equal to $144$ ?

• A. $3,15$
• B. $-3,-15$
• C. $-3,15$
• D. $3,-15$

Answer 1

Answer: (A), (B)

$-3,-15$

Given equation is $x^2+px+45=0$.
Let $a,b$ be the roots of the equation.
Then,
$a+b=-p$
$ab=45$
It is given that,
$(a-b{)}^2=144$
$(a+b{)}^2-4ab=144$
$\Rightarrow p^2-4\times 45=144$
$\Rightarrow p^2=324$
$\Rightarrow p=\pm 18$
Substituting $p=18$ in the given equation, we obtain
$x^2+18x+45=0$
$\Rightarrow (x+3)(x+15)=0$
$\Rightarrow x=-3,-15$
Substituting $p=-18$ in the given equation, we obtain
$x^2-18x+45=0$
$(x-3)(x-15)=0$
$\Rightarrow x=3,15$
Hence, the roots of the given equation are $-3,-15$ or $3,15$