Question

If the roots of the ploynomial $x^2=-x+q$ are double in value to the $2x^2-5x-3,$ find the values of $q.$

Answer 1

We have

$2x^2-5x-3=0$​

$\Rightarrow 2x^2-6x+x-3=0$​

$\Rightarrow 2x(x-3)+1(x-3)=0$​

$\Rightarrow (x-3)(2x+1)=0$​

$\Rightarrow x=3,-\frac{1}{2}$​

The roots of $x^2=-x+q$ are double the values of roots of $2x^2-5x-3=0$​

Since the roots of $2x^2-5x-3=0$ are $x=3,-\frac{1}{2}$​

Therefore, the roots of $x^2=-x+q$ are $x=6,-1$

Now, $x^2+x-q=0$

Product of roots $=\frac{\text{Constant term}}{\text{Coefficient of}{x}^2}$

$6\times (-1)=\frac{-q}{1}$

$\therefore q=6$