A quadratic equation has the form $ax^2 + bx + c = 0$. There are three main methods for solving them.
Method 1 — Factorising (fastest when it works): $$x^2 + 5x + 6 = 0 \Rightarrow (x+2)(x+3) = 0 \Rightarrow x = -2 \text{ or } x = -3$$
Method 2 — Quadratic formula (always works): $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Example: $2x^2 - 3x - 5 = 0$ $$x = \frac{3 \pm \sqrt{9 + 40}}{4} = \frac{3 \pm 7}{4} \Rightarrow x = 2.5 \text{ or } x = -1$$
Method 3 — Completing the square (useful for exact answers or the vertex): $$x^2 + 6x + 1 = 0 \Rightarrow (x+3)^2 - 8 = 0 \Rightarrow x = -3 \pm 2\sqrt{2}$$
The discriminant $b^2 - 4ac$ tells you how many solutions:
- $> 0$: two solutions
- $= 0$: one repeated solution
- $< 0$: no real solutions
Common error: forgetting the $\pm$ in the formula, giving only one solution.