Solve the equation $x^2 - 6x - 23 = 0$ - Leaving Cert Mathematics - Question 4 - 2013

To solve the quadratic equation $x^2 - 6x - 23 = 0$, we will use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = -6$, and $c = -23$. Let's calculate it step by step:

1. Calculate the discriminant: $b^2 - 4ac = (-6)^2 - 4(1)(-23) = 36 + 92 = 128$
2. Substitute into the quadratic formula: $x = \frac{6 \pm \sqrt{128}}{2}$
3. Simplify further: $x = 3 \pm 4 \sqrt{2}$

The solutions are therefore:

$x = 3 + 4\sqrt{2} \, \text{and} \, x = 3 - 4\sqrt{2}$