Find the values of the parameter $a$ not equal to zero for which

Question

Find the values of the parameter $a$ not equal to zero for which one of the roots of the quadratic equation $x^2-x-3a=0$ is double of one of the roots of the equation $x^2-x-a=0$

Answer 1

Let the root of $\,x^2-x-a=0\,$ be $\,b\,$, and $\,2b\,$ be the corresponding root of $\,x^2-x-3a = 0\,$:

$$ \begin{align} \begin{cases} b^2-b-a &= 0 \\ 4b^2 - 2b - 3a &= 0 \end{cases} \end{align} $$

Eliminating $\,b^2\,$ between the two gives:

$$ 2b + a = 0 \;\;\iff\;\; b = -\frac{a}{2} $$

Substituting back in either equation then gives:

$$ a^2-2a = 0 \;\;\iff\;\; a \in \{0, 2\} $$