9. (a) (i) Factorise $x^{2} + 7x - 30$ - Junior Cycle Mathematics - Question 9 - 2015

Using the factorised form found earlier, we can set each factor equal to zero:

$(x + 10) = 0 \quad \text{or} \quad (x - 3) = 0$

This gives us:

$x = -10 \quad \text{or} \quad x = 3.$

To solve this quadratic equation, we will use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 2$, $b = -7$, and $c = -10$.

1. Calculate the discriminant:

   $b^2 - 4ac = (-7)^2 - 4 \times 2 \times (-10) = 49 + 80 = 129.$

2. Apply the quadratic formula:

   $x = \frac{7 \pm \sqrt{129}}{4}.$

This results in two potential solutions:

$x \approx \frac{7 + \sqrt{129}}{4} \approx 4.59$

and

$x \approx \frac{7 - \sqrt{129}}{4} \approx -1.09.$

Thus, the answers are:

$x \approx 4.59 \quad \text{or} \quad x \approx -1.09$ (correct to two decimal places).