Solving Quadratic Equations: ax² + bx + c = 0
A quadratic equation is any equation of the form ax² + bx + c = 0, where a ≠ 0. Quadratics appear in physics (projectile motion), engineering (structural analysis), economics (optimisation), and countless other fields.
The Quadratic Formula
x = (−b ± √(b² − 4ac)) ÷ (2a)The Discriminant (b² − 4ac)
- Positive: Two distinct real roots
- Zero: One repeated real root (x = −b ÷ 2a)
- Negative: Two complex conjugate roots (no real solution)
Worked Example: 2x² − 5x − 3 = 0
a=2, b=−5, c=−3
Discriminant = 25 + 24 = 49
x = (5 ± 7) ÷ 4
x₁ = 12/4 = 3 | x₂ = −2/4 = −0.5Alternative: Factoring
When roots are integers, factoring is faster. x² − 5x + 6 = (x−2)(x−3) = 0 → x = 2 or 3. The quadratic formula always works when factoring is unclear.
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Real-World Applications
Quadratic equations describe a surprising range of real phenomena. In physics, projectile motion follows a parabolic path governed by h(t) = -½gt² + v₀t + h₀, a quadratic in time t. In economics, profit functions are often quadratic: P(x) = -ax² + bx - c, where x is units sold. Finding the profit-maximising output requires solving for the vertex (x = -b/2a). In engineering, stress distribution in beams can be quadratic. Architects use parabolic arches, which are quadratic curves, for structural efficiency. Even stopping distances in road safety modelling follow quadratic relationships between speed and distance.
Three Methods to Solve
- Quadratic formula: Always works. x = (-b ± √(b²-4ac)) / 2a. Use when factoring is not obvious.
- Factoring: Faster when roots are integers. x² - 5x + 6 = (x-2)(x-3) = 0, so x = 2 or 3. Works when you can spot factors quickly.
- Completing the square: Useful for deriving the quadratic formula and for converting to vertex form y = a(x-h)² + k, which directly gives the parabola's vertex at (h, k).
Frequently Asked Questions
What does a negative discriminant mean in practice?
When b²-4ac < 0, there are no real solutions — the parabola does not cross the x-axis. In applied problems this can mean "no solution exists" (e.g., a projectile never reaches that height) or that the model needs revisiting. Complex roots (involving √(-1) = i) arise and are meaningful in electrical engineering and signal processing.
How do I know which method to use?
Try factoring first if the coefficients are small integers. If the discriminant is a perfect square, factoring will work. Otherwise, use the quadratic formula. Completing the square is most useful when you need the vertex form for graphing or optimisation.
Can a quadratic equation have only one solution?
Yes. When the discriminant equals zero (b² - 4ac = 0), there is exactly one real solution (a repeated root): x = -b/2a. Geometrically, the parabola is tangent to the x-axis — it just touches without crossing.