Quadratic Equation Solver
Solve ax² + bx + c = 0 - find real and complex roots instantly.
x² 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, and c are real numbers and a ≠ 0. The term "quadratic" comes from the Latin word quadratus meaning square, because the highest power of the variable is 2 (squared).
Quadratic equations arise everywhere in mathematics, physics, and engineering. The trajectory of a thrown ball follows a parabolic path described by a quadratic equation. The area of a rectangle with a fixed perimeter is a quadratic function of its dimensions. In electronics, the resonant frequency of an LC circuit involves solving a quadratic. In finance, calculating the internal rate of return can require solving a quadratic. Even the perspective projection in 3D graphics uses quadratic equations.
The graph of y = ax² + bx + c is always a parabola. When a > 0, it opens upward (U-shape); when a < 0, it opens downward (∩-shape). The roots of the equation are the x-coordinates where the parabola crosses the x-axis. The vertex is the highest or lowest point of the parabola, and the axis of symmetry passes through it vertically.
There are four methods to solve quadratic equations: factoring (fast for simple cases), completing the square (foundational technique), the quadratic formula (always works), and graphing (visual but imprecise). The quadratic formula is derived by completing the square on the general form and gives exact solutions - including complex roots - for any quadratic equation.
📐 Quadratic Formula
The discriminant Δ = b² − 4ac determines how many real solutions exist:
The vertex of the parabola is at (h, k) where h = −b/(2a) and k = c − b²/(4a). The sum of the roots is −b/a and the product of the roots is c/a (Vieta's formulas).
📖 How to Use This Calculator
Steps to Solve a Quadratic Equation
💡 Example Calculations
Example 1 - Two real roots: x² − 5x + 6 = 0
Example 2 - One repeated root: x² − 6x + 9 = 0
Example 3 - Complex roots: x² + x + 1 = 0
❓ Frequently Asked Questions
🔗 Related Calculators
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b²−4ac)) / (2a). It gives the two roots of any quadratic equation ax² + bx + c = 0 where a ≠ 0. The ± sign means there are usually two solutions: one using + and one using −.
What is the discriminant and what does it tell you?
The discriminant is the expression b² − 4ac inside the square root of the quadratic formula. If discriminant > 0: two distinct real roots. If discriminant = 0: one repeated real root (a perfect square trinomial). If discriminant < 0: two complex (imaginary) conjugate roots, meaning the parabola does not cross the x-axis.
How do I solve a quadratic equation by factoring?
For simple cases, try to find two numbers that multiply to ac and add to b, then factor. For example, x² + 5x + 6 = 0: find numbers that multiply to 6 and add to 5 - those are 2 and 3. So (x+2)(x+3) = 0, giving roots x = −2 and x = −3. When factoring is not obvious, use the quadratic formula instead.
Can a quadratic equation have no real solutions?
Yes. When the discriminant (b²−4ac) is negative, the equation has no real solutions - instead, it has two complex conjugate roots of the form p ± qi where i = √(−1). Graphically, this means the parabola y = ax²+bx+c does not intersect the x-axis at all.
What is vertex form of a quadratic and how is it related?
Vertex form is y = a(x−h)² + k, where (h, k) is the vertex (turning point) of the parabola. h = −b/(2a) and k = c − b²/(4a). The axis of symmetry is x = h. The vertex form is useful for graphing and understanding transformations of the parabola.
What does the discriminant tell you about the roots?
The discriminant is D = b^2 - 4ac. It determines the nature of the roots: (1) D > 0: two distinct real roots (the parabola crosses the x-axis twice). (2) D = 0: one repeated real root (the parabola touches the x-axis at exactly one point - the vertex). (3) D < 0: two complex conjugate roots (the parabola does not cross the x-axis). The discriminant is calculated before using the full quadratic formula.
When should I use the quadratic formula vs factoring?
Use factoring when the coefficients are small integers and factors are easy to spot. Example: x^2 + 5x + 6 = (x+2)(x+3), so x = -2 or x = -3. Use the quadratic formula when: (1) the equation cannot be factored easily, (2) coefficients are large or involve fractions or decimals, (3) the roots are irrational or complex. The quadratic formula always works for any quadratic equation, making it the reliable general-purpose method.
What is the vertex of a parabola and how is it found?
The vertex is the highest or lowest point of the parabola y = ax^2 + bx + c. Vertex x-coordinate: x = -b / (2a). Vertex y-coordinate: substitute back into the equation. Example: for y = 2x^2 - 4x + 1: vertex x = -(-4) / (2 x 2) = 1. Vertex y = 2(1)^2 - 4(1) + 1 = -1. Vertex = (1, -1). If a > 0, the vertex is a minimum. If a < 0, the vertex is a maximum.