Quadratic Equation Solver
Solve any quadratic equation of the form ax² + bx + c = 0. Enter the coefficients a, b, and c to find real and complex roots, the discriminant, and the vertex form of the parabola.
How to Use This Quadratic Solver
Follow these simple steps to solve any quadratic equation:
- Enter the coefficient a (the x² term). Must be non-zero.
- Enter the coefficient b (the x term).
- Enter the coefficient c (the constant term).
- Click the "Solve" button to calculate the roots.
- Review the discriminant, roots, and vertex form in the results section.
Quadratic Formula
The quadratic equation ax² + bx + c = 0 is solved using the quadratic formula:
The discriminant D = b² - 4ac determines the nature of the roots:
- D > 0: Two distinct real roots
- D = 0: One repeated real root
- D < 0: Two complex conjugate roots
The vertex form of the parabola is: y = a(x - h)² + k, where h = -b/(2a) and k = f(h).
Examples
Equation: x² - 5x + 6 = 0 (a=1, b=-5, c=6)
D = (-5)² - 4(1)(6) = 25 - 24 = 1
x = (5 ± 1) / 2 → x₁ = 3, x₂ = 2
Equation: x² + 2x + 5 = 0 (a=1, b=2, c=5)
D = (2)² - 4(1)(5) = 4 - 20 = -16
x = (-2 ± 4i) / 2 → x = -1 ± 2i
Equation: x² - 6x + 9 = 0 (a=1, b=-6, c=9)
D = (-6)² - 4(1)(9) = 36 - 36 = 0
x = 6 / 2 → x = 3 (repeated)
Frequently Asked Questions
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 constants and a ≠ 0. It is called "quadratic" because the highest power of the variable x is 2. The graph of a quadratic equation is a parabola.
What does the discriminant tell you?
The discriminant (D = b² - 4ac) reveals the nature of the roots without solving the equation. If D is positive, there are two distinct real roots. If D is zero, there is exactly one repeated real root. If D is negative, the equation has two complex conjugate roots with no real solutions.
What is the difference between real and complex roots?
Real roots are numbers that appear on the number line (e.g., 2, -3.5). Complex roots involve the imaginary unit i where i² = -1, and appear as conjugate pairs like a ± bi. Complex roots occur when the discriminant is negative, meaning the parabola never crosses the x-axis.
Can a = 0 in a quadratic equation?
No. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0 because it divides by 2a. Our calculator validates that a is non-zero before computing.