What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. Quadratic equations appear throughout algebra, physics, engineering, and economics — from calculating projectile trajectories to modelling profit functions.
How do you solve a quadratic equation?
The most general method is the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Enter the three coefficients a, b, and c into the calculator and it returns both roots instantly. Alternative methods include factoring (when roots are rational), completing the square, and graphing. Factoring is fastest when the discriminant is a perfect square and the coefficients are small integers, while completing the square is preferred for converting to vertex form. The quadratic formula works for every quadratic equation, regardless of whether the roots are real or complex.
What is the quadratic formula?
The quadratic formula derives from completing the square on the general equation ax² + bx + c = 0. It gives two solutions:
- x₁ = (−b + √(b² − 4ac)) / 2a
- x₂ = (−b − √(b² − 4ac)) / 2a
The expression under the square root, D = b² − 4ac, is called the discriminant and determines the nature of the roots. For related calculations involving right triangles, see the Pythagorean theorem calculator.
What are some quadratic equation examples?
Two real roots: x² − 5x + 6 = 0 → D = 25 − 24 = 1 → x₁ = (5 + 1)/2 = 3, x₂ = (5 − 1)/2 = 2.
One repeated root: x² − 6x + 9 = 0 → D = 36 − 36 = 0 → x = 6/2 = 3.
Complex roots: x² + 2x + 5 = 0 → D = 4 − 20 = −16 → x = (−2 ± 4i)/2 = −1 ± 2i.
What does the discriminant tell you?
The discriminant D = b² − 4ac reveals the nature of the roots without solving the equation:
- D > 0 — two distinct real roots (the parabola crosses the x-axis twice)
- D = 0 — one repeated real root (the parabola touches the x-axis at its vertex)
- D < 0 — two complex conjugate roots (the parabola does not cross the x-axis)
If D is a perfect square and the coefficients are integers, the roots are rational and the equation can be factored. This is useful when working with fractions or simplifying algebraic expressions.
What is the vertex of a parabola?
Every quadratic equation y = ax² + bx + c graphs as a parabola with a vertex at (h, k) where h = −b/(2a) and k = c − b²/(4a). The vertex is the minimum point when a > 0 and the maximum when a < 0. The axis of symmetry passes through the vertex at x = h. Converting to vertex form gives y = a(x − h)² + k, which makes graphing straightforward. For other geometric calculations, explore the triangle calculator.
How is the quadratic equation used in real life?
Quadratic equations model any situation where a quantity depends on the square of a variable. Physicists use them to calculate projectile height: h = −½gt² + v₀t + h₀. Engineers size structural arches using parabolic curves. Economists find maximum profit by solving for the vertex of a revenue function. Even everyday tasks — like finding the dimensions of a garden with a fixed perimeter and maximum area — reduce to quadratic equations. If the equation involves percentages, the percentage calculator can help with related conversions.