The standard form
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola. There are four main ways to solve it, and you should pick the quickest for the problem in front of you.
1. Factoring
If the expression factors, use the zero product property: if a product is zero, one factor is zero.
x² − 7x + 12 = 0 factors as (x − 3)(x − 4) = 0, so x = 3 or x = 4. To factor x² + bx + c, look for two numbers that multiply to c and add to b (here 3 × 4 = 12 and 3 + 4 = 7, then both get minus signs).
2. Square roots
If there is no middle term, isolate the square: x² = 49 gives x = 7 or x = −7. Always keep both signs.
3. Completing the square
Rewrite x² + bx as (x + b/2)² − (b/2)². To complete the square for x² + 6x you add 9, because (6/2)² = 9. This method also turns a quadratic into vertex form y = a(x − h)² + k, whose vertex is (h, k).
4. The quadratic formula
For any quadratic:
x = (−b ± √(b² − 4ac)) / (2a)
The part under the root, b² − 4ac, is the discriminant:
- Positive: two different real solutions.
- Zero: one repeated real solution.
- Negative: no real solutions (two complex ones).
Worked example
Solve 2x² + 3x − 5 = 0.
Here a = 2, b = 3, c = −5.
- Discriminant: 3² − 4(2)(−5) = 9 + 40 = 49, whose square root is 7.
- x = (−3 ± 7) / 4.
- So x = 4/4 = 1, or x = −10/4 = −5/2.
Check x = 1: 2 + 3 − 5 = 0. ✓.
A shortcut: the two solutions add to −b/a and multiply to c/a. Here 1 + (−5/2) = −3/2, which is −3/2. ✓.
Reading a parabola
For y = ax² + bx + c the axis of symmetry is x = −b/(2a). The vertex lies on that line; substitute to find its y-value. If a > 0 the parabola opens up and the vertex is a minimum; if a < 0 it opens down and the vertex is a maximum. For y = x² − 6x + 1 the axis is x = 3, and y(3) = 9 − 18 + 1 = −8, so the vertex is (3, −8).
Common mistakes
- Dropping the ± when taking a square root.
- Forgetting to move everything to one side (= 0) before factoring.
- Sign slips with b: remember the formula starts with −b.