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  4. If the roots of the equation x2 + px + c = 0 are 2, -2 and the x2 + bx + q = 0 are -1, -2, then the roots of the equation x2 +bx + c = 0 are

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Q. If the roots of the equation $x^2 + px + c = 0$ are $2, -2$ and the $x^2 + bx + q = 0$ are $-1, -2$, then the roots of the equation $x^2 +bx + c = 0$ are

KEAMKEAM 2013Complex Numbers and Quadratic Equations

Solution:

Given, roots of the equation $x^{2}+p x+c=0$ are 2 and $-2$.
$\therefore \,-p=2-2$
$ \Rightarrow \,p=0$
and $(2) \cdot(-2)=c $
$ \Rightarrow \, c=-4$
Again, roots of the equation $x^{2}+b x+q=0$ are $-1$ and $-2$
$\therefore \, -b=-1-2$
$ \Rightarrow \,b=3$
and $(-1) \cdot(-2)=q$
$ \Rightarrow \, q=2$
$\therefore \, x^{2}+b x+c \equiv x^{3}+3 x-4=0$
$\Rightarrow \, (x-1)(x+4)=0$
So, the required roots are $1$ and $-4$