Quadratic Equation Solver
Enter coefficients a, b, c to find the roots of any quadratic equation using the quadratic formula.
The Quadratic Formula
For any quadratic equation ax² + bx + c = 0 (where a ≠ 0), the roots are given by:
Two distinct real roots. The parabola crosses the x-axis at two different points.
One repeated real root (equal roots). The parabola just touches the x-axis at one point (vertex).
Two complex (imaginary) roots. The parabola does not cross the x-axis at all.
Frequently asked questions
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a, b, c are constants and a ≠ 0. The highest power of the variable x is 2 (squared). Examples: x² - 5x + 6 = 0, 2x² + 3x - 2 = 0.
What is the discriminant?
The discriminant D = b² - 4ac is a part of the quadratic formula under the square root. Its value tells you the nature of the roots without fully solving the equation. D > 0: two real roots; D = 0: one repeated real root; D < 0: two complex roots. Useful for quickly determining root type in competitive exams.
What are other methods to solve quadratic equations?
Besides the quadratic formula, quadratics can be solved by: (1) Factoring — expressing as (x-p)(x-q) = 0; (2) Completing the square — converting to vertex form; (3) Graphing — finding x-intercepts. The quadratic formula always works; factoring is faster when factors are obvious.
What are the sum and product of roots?
For ax² + bx + c = 0 with roots α and β: Sum of roots = α + β = -b/a; Product of roots = α × β = c/a. These Vieta's formulas let you find properties of roots without solving. Example: for x² - 5x + 6 = 0, sum = 5, product = 6, giving roots 2 and 3.
Where are quadratic equations used in real life?
Quadratics appear in physics (projectile motion: h = ut - ½gt²), engineering (parabolic bridges, satellite dishes), finance (maximum profit calculations), and computer graphics (bezier curves). The shape of a thrown ball's path and the design of a parabolic antenna both involve quadratic equations.