Quadratic Formula
Finds the roots of any quadratic equation ax² + bx + c = 0.
Formula
Variables
Example
For 2x² + 5x - 3 = 0: x = 0.5 or x = -3
How it works
This formula gives the x-values where a quadratic graph crosses the x-axis, so it solves ax² + bx + c = 0 for any nonzero a. It comes from completing the square on the general quadratic, which rearranges the equation into a form where x can be isolated exactly. The expression inside the square root, b² - 4ac, tells you whether the equation has two real roots, one repeated root, or no real roots.
When to use it
Use it whenever you need exact solutions to a quadratic equation and factoring is difficult or impossible. It is especially useful in algebra, graphing parabolas, projectile-motion problems, and any situation where a quantity is modeled by a second-degree polynomial.
Worked example, step by step
Solve 3x² - 4x - 4 = 0 Step 1: Identify a = 3, b = -4, c = -4 Step 2: Substitute into the quadratic formula: x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-4)}}{2(3)} Step 3: Simplify inside the square root: x = \frac{4 \pm \sqrt{16 + 48}}{6} x = \frac{4 \pm \sqrt{64}}{6} Step 4: Compute the two roots: x = \frac{4 + 8}{6} = 2 x = \frac{4 - 8}{6} = -\frac{2}{3} Final answer: x = 2 or x = -2/3
Common mistakes to avoid
- Using the wrong sign for b when substituting into -b
- Forgetting to divide the entire numerator by 2a
- Stopping before simplifying the square root and checking both ± solutions
Did You Know?
Babylonians were solving quadratic equations on clay tablets nearly 4,000 years ago — long before algebraic notation existed.
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