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Solve the equations

You should find a total of $5$ solutions. Find the product of these $5$ solutions and give your answer below.

Hint

For each equation, you need to get it in the form

x^{2} + b x + c = 0

first.

Hint

For (b), recall that $30 x^{- 1}$ means $\frac{30}{x}$ , so we have

x = \frac{30}{x} - 1

You should multiply both sides by $x$ to simplify first.

Solution

First we need to expand out the left-hand side
$\begin{aligned} - x^{2} + 4 x + 21 & = 25 \\ x^{2} - 4 x + 4 & = 0 \\ (x - 2)^{2} & = 0 \\ x & = 2 \end{aligned}$
We must multiply by $x$ first
$\begin{aligned} x & = \frac{30}{x} - 1 \\ x^{2} & = 30 - x \\ x^{2} + x - 30 & = 0 \\ (x + 6) (x - 5) & = 0 \\ x & \in {- 6, 5} \end{aligned}$
Again, the key is to expland and simplify:
$\begin{aligned} (2 x - 1) (x + 3) & = 7 + 2 x + x^{2} \\ 2 x^{2} + 5 x - 3 & = 7 + 2 x + x^{2} \\ x^{2} + 3 x - 10 & = 0 \\ (x + 5) (x - 2) & = 0 \\ x & \in {- 5, 2} \end{aligned}$