The quadratic formula. That long square-root-containing expression from secondary school maths that reliably caused confusion. The good news: it's not actually hard once someone explains it properly. And it solves problems that appear in physics, engineering, economics, and construction — anywhere a relationship involves a squared term.
What Is a Quadratic Equation?
Any equation in the form ax² + bx + c = 0 where a ≠ 0. The squared term (x²) makes it "quadratic." Examples: x² − 5x + 6 = 0 or 2x² + 3x − 2 = 0. The solutions (called roots) are x-values that make the equation true. For related arithmetic our percentage calculator handles the simpler linear maths; for quadratics, use the formula below.
The Formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The ± gives two solutions — one with addition, one with subtraction. A parabola (the curve a quadratic describes) typically crosses the x-axis at two points. Check your working with our grade calculator when applying this to academic problems.
Step-by-Step Example
Solve: x² − 5x + 6 = 0. Here a=1, b=−5, c=6.
- b² = (−5)² = 25
- 4ac = 4 × 1 × 6 = 24
- Discriminant (b² − 4ac) = 25 − 24 = 1
- √1 = 1
- x = (5 + 1) ÷ 2 = 3 and x = (5 − 1) ÷ 2 = 2
Check: 3² − 15 + 6 = 0 ✓ and 2² − 10 + 6 = 0 ✓
The Discriminant Tells You How Many Solutions
- b² − 4ac > 0: two distinct real solutions (parabola crosses x-axis twice)
- b² − 4ac = 0: one repeated solution (parabola just touches x-axis)
- b² − 4ac < 0: no real solutions (parabola doesn't reach x-axis)
When to Use the Formula
Three methods exist: factoring, completing the square, and the quadratic formula. The formula always works — the others only work in certain cases. When in doubt, always use the formula. It's slightly slower but never fails.
Real-World Applications
- Projectile motion: "when does the ball land?" involves a quadratic (gravity creates the squared term)
- Revenue optimisation: finding the price that maximises revenue when demand changes with price
- Construction: area constraints with a path of uniform width around a garden produce quadratics
- Physics: kinetic energy, distance under constant acceleration
A ball thrown upward: h = −5t² + 20t. When h=0? Solutions: t=0 (thrown) and t=4 seconds (lands). Clean, precise, and genuinely useful.
Further reading: Khan Academy's comprehensive quadratic unit includes worked examples and practice. Practice quadratic equations at Khan Academy.