About the Quadratic Equation Solver
Quadratic Equation Solver takes the three coefficients a, b and c of ax² + bx + c = 0 and shows everything about the equation. You get the two roots, one repeated root, or a pair of complex roots, together with the discriminant, the vertex, the axis of symmetry, the y-intercept and, when the roots are whole numbers or simple fractions, the factored form. A steps panel shows the quadratic formula filled in with your numbers, and a small graph plots the parabola with its roots and vertex. The roots come from a numerically stable version of the formula, and the results are rounded to six decimal places, so roots that are irrational are approximations. If a is 0, the equation is not quadratic and the tool solves the straight-line equation instead.
Updated October 2026
How to use the Quadratic Equation Solver
- 1Type the coefficients a, b and c. Decimals and negative numbers are allowed.
- 2Read the roots in the result box and the equation as it is written.
- 3Check the discriminant, vertex, axis, y-intercept and factored form in the table.
- 4Open Show steps to see the quadratic formula with your numbers.
- 5Look at the graph to see where the parabola crosses the x-axis.
The quadratic formula
Any equation that can be written as ax² + bx + c = 0 with a not equal to zero can be solved with the formula x = (−b ± √(b² − 4ac)) ÷ 2a. The plus and the minus give the two roots.
When the two roots are very different in size, the plain formula can lose accuracy because it subtracts two nearly equal numbers. This solver first computes the larger root safely and gets the other from c divided by it, so a case like x² − 100,000,000x + 1 = 0 still gives 0.00000001 for the small root.
What the discriminant tells you
- D > 0: two different real roots, and the parabola crosses the x-axis twice.
- D = 0: one repeated root, and the vertex lies on the x-axis.
- D < 0: no real roots; the graph never touches the x-axis, and the roots are complex.
- The tool treats a discriminant that is tiny compared with the coefficients as zero, to avoid rounding noise.
Factoring, completing the square and the formula
Factoring is quickest when the roots are simple numbers: look for two numbers that multiply to c and add to b when a is 1. Completing the square rewrites the equation as a perfect square plus a constant and is how the formula is derived. The formula works for every quadratic, so it is the safe choice when neither of the others is obvious.
The factored form appears in the results only when both roots are whole numbers or fractions with a denominator up to 100. For other equations, use the roots directly.
Reading the graph
The curve is the graph of y = ax² + bx + c. Dots mark the points where it crosses the x-axis, which are the real roots, and the ring marks the vertex. A positive a opens the parabola upwards and a negative a opens it downwards.
The picture is scaled to include the roots and the vertex, so it is a sketch, not a plot with numbered axes. For a general sum with more than one operation, use Online Calculator, and for fractions in your answers use Fraction Calculator. For a quick sum use the Online Calculator, and for a mean of several answers the Average Calculator.
Features
- Real, repeated and complex roots
- Discriminant, vertex, axis of symmetry and y-intercept
- Factored form for rational roots
- Steps with the quadratic formula
- Graph of the parabola drawn in the page
- Handles a = 0 as a straight-line equation
- Runs in your browser
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the root, b² − 4ac, is the discriminant. Type your a, b and c above and the steps panel shows the formula with your numbers in place.
What does the discriminant tell me?
If it is positive, the equation has two different real roots. If it is zero, there is one repeated root, and the parabola just touches the x-axis. If it is negative, there are no real roots and the solutions are a pair of complex numbers.
What if a is zero?
Then the x² term disappears and the equation is a straight line, bx + c = 0, with the single solution x = −c ÷ b. The calculator says so and gives that answer. If b is also 0, there is either no solution or every number is a solution, depending on c.
How do I solve a quadratic with complex roots?
When the discriminant is negative, write the roots as x = −b ÷ 2a ± (√(−D) ÷ 2a) i, where i is the square root of −1. For x² + x + 1 = 0, the roots are −0.5 ± 0.866025i. The tool shows them in that form.
How do I factor a quadratic?
Find the roots r and s, then write a(x − r)(x − s). For x² − 3x + 2 the roots are 1 and 2, so it factors as (x − 1)(x − 2). The tool shows this form when both roots are whole numbers or simple fractions.
What is the vertex of a parabola?
It is the turning point, the lowest point when a is positive and the highest when a is negative. Its x-value is −b ÷ 2a and its y-value comes from putting that x back into the equation. The axis of symmetry is the vertical line through it.