The quadratic formula solves any quadratic equation of the form $ax^2 + bx + c = 0$, including those that cannot be factorised.
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Steps:
- Rearrange into the form $ax^2 + bx + c = 0$ (zero on the right)
- Identify $a$, $b$, and $c$ — include negative signs
- Substitute into the formula
- Calculate the discriminant $b^2 - 4ac$ first, then take the square root
- Apply $+$ and $-$ separately to get two solutions
- Round to the required degree of accuracy (usually 1 or 2 decimal places)
Example: Solve $2x^2 - 3x - 5 = 0$. Here $a = 2$, $b = -3$, $c = -5$. $$x = \frac{3 \pm \sqrt{9 + 40}}{4} = \frac{3 \pm 7}{4}$$ $$x = \frac{10}{4} = 2.5 \quad \text{or} \quad x = \frac{-4}{4} = -1$$
Discriminant: if $b^2 - 4ac < 0$, there are no real solutions. If $= 0$, exactly one solution.
Common error: forgetting to write $\pm$ and only finding one solution, or dropping the negative sign on $b$ when substituting.