Solve ax² + bx + c = 0
This quadratic equation calculator solves ax² + bx + c = 0 and shows every step. Enter the coefficients a, b and c, then hit Solve — or try a preset.
Step-by-step solution
Parabola graph
A live quadratic equation grapher — the parabola redraws every time you solve.
The quadratic formula, step by step
Every quadratic equation ax² + bx + c = 0 (with a ≠ 0) is solved by the quadratic formula:
x = (−b ± √(b² − 4ac)) / 2a
Think of this page as a quadratic formula calculator with steps: plug in a, b and c, evaluate the discriminant under the root, then take the plus and minus branches. Unlike factoring, the formula works for every quadratic — messy fractions, irrational roots and complex roots included.
The discriminant: b² − 4ac
The discriminant formula is D = b² − 4ac — compute it first, before the full quadratic formula.
| Discriminant | Roots | What the graph does |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola crosses the x-axis twice |
| D = 0 | One repeated (double) root, x = −b/2a | Parabola just touches the x-axis at the vertex |
| D < 0 | Two complex conjugate roots | Parabola never touches the x-axis |
Three ways to solve
1. Factoring — fastest when it works. Write the quadratic as a product of two binomials, e.g. x² − 5x + 6 = (x − 2)(x − 3), then each factor gives a root. Best for small integer coefficients.
2. Quadratic formula — the universal method above. Always works.
3. Completing the square — follow these steps: rewrite the equation as a perfect square: (x + b/2a)² = (b² − 4ac)/4a², then take square roots. This is also how the quadratic formula itself is derived.
Three forms of a quadratic
Standard form y = ax² + bx + c —
best for the discriminant and the y-intercept (which is just c).
Vertex form y = a(x − h)² + k —
exposes the vertex (h, k) directly. The vertex formula gives h = −b/2a and k = c − b²/4a, so the vertex form formula is fully determined by a, b and c.
Factored form y = a(x − r₁)(x − r₂) —
exposes the roots r₁ and r₂ (the x-intercepts).
What this tool does not do
- It solves quadratic equations in one variable (ax² + bx + c = 0) only — not cubic, exponential or logarithmic equations.
- It is not quadratic regression (fitting a parabola through data points — a statistics problem).
- If you enter a = 0, the equation becomes linear (bx + c = 0); the tool says so and solves that instead of pretending.
Worked examples with answers
- Discriminant: D = 25 − 24 = 1
- Formula: x = (5 ± 1) / 2
- Factoring check: (x − 2)(x − 3) ✓
- Discriminant: D = 4 − 20 = −16 (no real roots)
- Formula: x = (−2 ± 4i) / 2
- Vertex form: (x + 1)² + 4
- Discriminant: D = 16 − 16 = 0 (double root)
- Root: x = −4 / 4 = −1
- Perfect square: 2(x + 1)² = 0
Frequently asked questions
How do I solve a quadratic equation step by step?
Start from standard form ax² + bx + c = 0. Step 1: compute the discriminant D = b² − 4ac. Step 2: if D is a perfect square or the numbers are friendly, try factoring; otherwise use the quadratic formula x = (−b ± √D) / 2a. Step 3: simplify the two roots. Step 4 (optional): verify by plugging each root back into the equation. The solver above does all four steps and shows its working.
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It gives the roots of any quadratic equation ax² + bx + c = 0 directly from the coefficients a, b, and c. It works for every quadratic — including ones with complex (imaginary) roots — which is why it is the most reliable method when factoring is not obvious.
What is the discriminant, and what does it tell you?
The discriminant is the part under the square root in the quadratic formula: D = b² − 4ac. It tells you the nature of the roots before you finish solving: D > 0 means two distinct real roots (the parabola crosses the x-axis twice), D = 0 means one repeated real root (the parabola just touches the x-axis at the vertex), and D < 0 means two complex conjugate roots (the parabola never touches the x-axis).
Can a quadratic equation have no real roots?
Yes. When the discriminant is negative (D < 0), the equation has no real roots — only two complex conjugate roots of the form p ± qi. On the graph this means the parabola floats entirely above the x-axis (if a > 0) or below it (if a < 0). A quadratic always has exactly two roots if you count complex numbers and multiplicity.
How do you find the vertex of a parabola?
For y = ax² + bx + c, the vertex is at x = −b / 2a. Plug that x back into the equation to get y = c − b² / 4a. The vertex is the parabola's minimum point when a > 0 (opens upward) and its maximum point when a < 0 (opens downward). The vertical line x = −b / 2a is the axis of symmetry.
When should I use the quadratic formula instead of factoring?
Try factoring first when a, b, and c are small integers — it is fast when it works (for example x² − 5x + 6 = (x − 2)(x − 3)). Use the quadratic formula when factoring is not obvious, when coefficients are large or fractional, or when the discriminant is not a perfect square. The formula never fails, so for homework and exams it is the safe default.
How do you complete the square?
To complete the square for ax² + bx + c = 0: divide by a (if a ≠ 1), move c to the other side, then add (b/2a)² to both sides so the left side becomes a perfect square: (x + b/2a)² = (b/2a)² − c/a. Take the square root of both sides and solve for x. For example, x² − 5x + 6 = 0 becomes (x − 5/2)² = 1/4, so x = 5/2 ± 1/2, giving x = 3 and x = 2.
What is vertex form, and how do I convert standard form to vertex form?
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex. To convert from standard form y = ax² + bx + c, complete the square: h = −b / 2a and k = c − b² / 4a. For example, x² − 5x + 6 becomes (x − 5/2)² − 1/4, revealing the vertex (5/2, −1/4) immediately.
How do you factor a quadratic when a is not 1?
Use the grouping method (also called the ac method): for ax² + bx + c, find two numbers that multiply to a·c and add to b. Split the middle term with those numbers, then factor by grouping. For 2x² − 3x − 2, a·c = −4; the numbers 1 and −4 multiply to −4 and add to −3, so 2x² + x − 4x − 2 = x(2x + 1) − 2(2x + 1) = (x − 2)(2x + 1).
What does it mean when the discriminant is zero?
A discriminant of zero means the quadratic has one repeated (double) real root: x = −b / 2a. The parabola touches the x-axis exactly once, at its vertex. Such quadratics are perfect squares — for example 2x² + 4x + 2 = 0 has D = 0 and the double root x = −1, and it factors as 2(x + 1)² = 0.
How do I find the x-intercepts and y-intercept of a quadratic?
The y-intercept is simply c — set x = 0 in y = ax² + bx + c. The x-intercepts are the real roots: solve ax² + bx + c = 0. If the discriminant is positive you get two x-intercepts; if zero, one (the vertex sits on the axis); if negative, there are no x-intercepts and the graph never crosses the x-axis.
Why does the quadratic formula work?
The quadratic formula is just completing the square written in general form. Starting from ax² + bx + c = 0, divide by a, complete the square to get (x + b/2a)² = (b² − 4ac) / 4a², then take square roots and isolate x — the result is x = (−b ± √(b² − 4ac)) / 2a. It is a proof you can replay on any specific equation.
What is the difference between a quadratic equation solver and quadratic regression?
A quadratic equation solver finds the x-values that make ax² + bx + c equal zero. Quadratic regression is statistics: it fits the best parabola through a set of data points. They share the word 'quadratic' but solve completely different problems — this tool only solves equations, it does not fit curves to data.
Can this tool solve linear equations too?
Yes, as a fallback: if you enter a = 0, the equation is no longer quadratic, so the tool solves the linear equation bx + c = 0 (giving x = −c/b) and tells you it did so. If a = 0 and b = 0 as well, there is nothing to solve and the tool says so instead of guessing.
How do I read the parabola graph: opening direction and axis of symmetry?
The sign of a controls the opening: a > 0 opens upward (∪, vertex is the minimum), a < 0 opens downward (∩, vertex is the maximum). The axis of symmetry is the vertical line x = −b / 2a through the vertex — the parabola mirrors across it. The graph above marks the vertex, the axis, and both intercepts so you can read all of this at a glance.
What are the three forms of a quadratic equation?
Standard form y = ax² + bx + c is best for computing the discriminant and y-intercept. Vertex form y = a(x − h)² + k exposes the vertex (h, k) directly. Factored (intercept) form y = a(x − r₁)(x − r₂) exposes the roots r₁ and r₂. The solver above shows all three forms for your equation so you can pick whichever the problem needs.