Quadratic Formula
Finds the roots of any quadratic equation ax² + bx + c = 0.
x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
1What it means
The quadratic formula gives the two solutions (roots) of a quadratic equation. The expression under the root, b² − 4ac, is called the discriminant: positive means two real roots, zero means one repeated root, and negative means no real roots.
2Variables
| Symbol | Meaning |
|---|---|
| a | Coefficient of x² (a ≠ 0) |
| b | Coefficient of x |
| c | Constant term |
| x | The unknown being solved for |
3Worked examples
Example 1 Worked solution
Q. Solve 2x² + 5x − 3 = 0.
- Here a = 2, b = 5, c = −3.
- Discriminant = b² − 4ac = 25 + 24 = 49.
- √49 = 7, so x = (−5 ± 7) / (2 × 2).
- x = 2/4 = 0.5 or x = −12/4 = −3.
✓ x = 0.5 or x = −3
4Where it's used
- ✦Solving projectile and motion problems where height is a quadratic in time.
- ✦Finding break-even points in profit/loss and cost problems.
- ✦Any word problem that reduces to a second-degree equation.
5Tips & common mistakes
- !Always write the equation as ax² + bx + c = 0 before reading off a, b, c.
- !Watch the sign of c — a common mistake is dropping the minus sign.
- !If the discriminant is negative, there are no real roots.