Standard form
A quadratic equation is any equation that can be written as:
ax² + bx + c = 0, where a ≠ 0
The discriminant
The discriminant tells you how many real roots an equation has, without solving it:
D = b² − 4ac
- If D > 0 → two distinct real roots
- If D = 0 → one repeated real root
- If D < 0 → no real roots (two complex roots)
The quadratic formula
For any quadratic in standard form:
x = (−b ± √D) / 2a
Worked example
Solve 2x² − 4x − 6 = 0.
Here a = 2, b = −4, c = −6.
D = (−4)² − 4(2)(−6) = 16 + 48 = 64
x = (4 ± 8) / 4 → x = 3 or x = −1
Completing the square
Rewrite ax² + bx + c in the form a(x − h)² + k. This is the same idea behind deriving the quadratic formula, and it's useful for finding the vertex of a parabola quickly.
Common exam mistakes
- Forgetting the ± sign when taking a square root
- Sign errors when substituting negative values of b or c
- Not simplifying the surd fully in the final answer
Practice
Try solving: x² − 5x + 6 = 0, 3x² + x − 2 = 0, and x² + 4x + 4 = 0. Check your answers by substituting back into the original equation.