Question

Find the quadratic equation whose roots are the additive inverses of the roots of the equation $x^2-5x+6=0$.

Answer 1

First we have to find the roots of the equation $x^2-5x+6=0$
$x^2-5x+6$
$\Rightarrow x^2-2x-3x+6=x(x-2)-3(x-2)\Rightarrow (x-2)(x-3)=0\Rightarrow (x-2)=0or(x-3)=0\Rightarrow x=2,3$
Now the roots of the required equation are the additive inverses of 2,3. i.e. -2,-3. So the required equation is
$\Rightarrow (x-(-2\left)\right)(x-(-3\left)\right)=0\Rightarrow (x+2)(x+3)=0\Rightarrow x^2+5x+6=0$
Therefore the required equation is $x^2+5x+6=0$.