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Solve the equations

The sum of the squares of the solutions to (a) and (b) is equal so $S$ .

Give the value of

16 S

Hint

For each equation, it is possible to factorise out an $x$ (and maybe also a common factor out of the coefficients).

Hint

For example, the first equation can be written as

2 x (x^{2} - x - 6) = 0

Hint

Each equation has exactly three solutions (you can't divide by $x$ )

Solution

We begin by factoring out $2 x$ :
$\begin{aligned} 2 x (x^{2} - x - 6) & = 0 \\ 2 x (x - 3) (x + 2) & = 0 \\ x & \in {- 2, 0, 3} \end{aligned}$
Note that $x = 0$ is a solution.
We factor out $3 x$ at the beginning:
$\begin{aligned} 3 x (2 x^{2} - x - 15) & = 0 \\ 3 x \frac{(2 x - 6)}{2} (2 x + 5) & = 0 \\ 3 x (x - 3) (2 x + 5) & = 0 \\ x & \in {- \frac{5}{2}, 0, 3} \end{aligned}$