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Enter a, b, and c to solve the quadratic equation and graph its parabola.
What Is a Quadratic Formula Calculator?
A quadratic formula calculator solves equations written as ax² + bx + c = 0, where a, b, and c are numerical coefficients and a is not zero. It substitutes those coefficients into x = (-b ± √(b² - 4ac)) / 2a, classifies the solutions, and shows the algebra that leads to each root.
Unlike factoring, the quadratic formula works for every genuine quadratic equation. The answers may be two distinct real numbers, one repeated real number, or a pair of complex conjugates. This page also calculates the vertex and axis of symmetry so the equation and its parabola can be understood together.
How the Discriminant Changes the Roots
The discriminant D = b² - 4ac controls how a parabola meets the x-axis. These three real charts show each possible case rather than using an empty decorative panel.
D > 0
Two real roots
D = 0
One repeated root
D < 0
Two complex roots
How to Use the Quadratic Formula
First place the equation in standard form with zero on one side. Identify a as the coefficient of x², b as the coefficient of x, and c as the constant. Preserve negative signs: in x² - 5x + 6 = 0, the values are a = 1, b = -5, and c = 6.
Calculate the discriminant, substitute all three coefficients into the formula, and simplify both the plus and minus branches. For this example D = 25 - 24 = 1, so x = (5 ± 1) / 2. The two roots are 2 and 3.
What the Discriminant Tells You
A positive discriminant produces two distinct real roots and two x-axis crossings. A zero discriminant produces one repeated root where the parabola touches the x-axis at its vertex. A negative discriminant produces no real crossing and two complex conjugate solutions.
The discriminant predicts the solution type before the square root is evaluated. It does not by itself give the roots, so the remaining quadratic-formula steps are still required.
How to Find the Vertex of a Parabola
The vertex x-coordinate is h = -b / 2a. Substitute h into y = ax² + bx + c to find the vertex y-coordinate k. The vertical line x = h is the axis of symmetry, and it divides the parabola into mirror-image halves.
When a is positive, the parabola opens upward and the vertex is its minimum. When a is negative, it opens downward and the vertex is its maximum. The constant c is the y-intercept because substituting x = 0 leaves y = c.
Real Roots, Repeated Roots, and Complex Roots
Two real roots
When D is positive, the square root term is real and nonzero. The plus and minus branches give different x-values.
Repeated root
When D equals zero, the square-root term disappears. Both branches reduce to x = -b / 2a.
Complex roots
When D is negative, √D contains i. The two answers have equal real parts and opposite imaginary parts.
Quadratic Formula Examples
These examples cover every discriminant class and include upward and downward parabolas.
| Equation | D | Root type | Solutions |
|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | Two real | 2, 3 |
| x² - 6x + 9 = 0 | 0 | Repeated | 3 |
| x² + 2x + 5 = 0 | -16 | Complex | -1 ± 2i |
| 2x² + 9x - 5 = 0 | 121 | Two real | -5, 0.5 |
| -x² + 4x + 5 = 0 | 36 | Two real | -1, 5 |
Quadratic Formula Calculator FAQ
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves every equation in the form ax² + bx + c = 0 when a is nonzero.
How do I identify a, b, and c in a quadratic equation?
Arrange the equation in descending powers with zero on the right. Read a from the x² term, b from the x term, and c from the constant, including each sign.
What does the discriminant tell you?
The discriminant is b² - 4ac. Its sign tells whether there are two real roots, one repeated root, or two complex conjugate roots.
Can the quadratic formula give complex roots?
Yes. A negative discriminant creates a square root of a negative number, which is expressed using i. The resulting roots are complex conjugates.
Why can coefficient a not equal zero?
If a equals zero, the x² term disappears and the equation is linear rather than quadratic. The quadratic formula divides by 2a, so it is not defined in that case.
How do I find the vertex from a, b, and c?
Calculate h = -b / 2a, then evaluate the polynomial at h to find k. The vertex is (h, k), and x = h is the axis of symmetry.
How do I check a quadratic formula answer?
Substitute each real root into the original polynomial. A correct root makes ax² + bx + c equal zero, allowing for small rounding residuals in decimal calculations.
When should I use factoring instead of the quadratic formula?
Factoring can be faster when integer factors are obvious. The quadratic formula is more general and remains reliable when factors are irrational, decimal, or complex.
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