Quadratic Equation Solver
Solve any quadratic equation of the form ax² + bx + c = 0. Enter the coefficients below and get instant results including real or complex roots, the discriminant, vertex, and axis of symmetry.
This tool also provides a step-by-step breakdown of the solution so you can follow the math and verify each calculation.
About the Quadratic Equation Solver
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. These equations appear throughout algebra, physics, engineering, and finance.
The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, provides a universal method for finding the roots of any quadratic equation. The discriminant (b² - 4ac) determines the nature of the roots: two real roots, one repeated root, or two complex conjugate roots.
Beyond finding roots, this solver also calculates the vertex and axis of symmetry of the corresponding parabola, which are essential for graphing and optimization problems.
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It gives the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0, where a ≠ 0. You simply substitute the coefficients a, b, and c into the formula to find x.
What does the discriminant tell you?
The discriminant is the expression b² - 4ac found under the square root in the quadratic formula. If it is positive, the equation has two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If it is negative, the equation has two complex conjugate roots with imaginary parts.
What are complex roots?
Complex roots occur when the discriminant (b² - 4ac) is negative, meaning you need to take the square root of a negative number. They are expressed in the form a + bi and a - bi, where i is the imaginary unit equal to √(-1). Complex roots always come in conjugate pairs.
How do you find the vertex of a parabola?
The vertex of the parabola defined by y = ax² + bx + c is located at the point (-b/(2a), f(-b/(2a))). First calculate the x-coordinate as -b/(2a), then substitute that value back into the equation to get the y-coordinate. The vertex represents the minimum point if a > 0, or the maximum point if a < 0.