Quadratic Equation Solver
Solve ax²+bx+c=0 with steps
🔒 In-browser · Method · · Disclaimer · Quadratic Equation Solver
Working out how to solve ax²+bx+c=0 with steps is easier with Quadratic Equation Solver, which answers as you type. You supply a, b and c to Quadratic Equation; the answer follows at once. Quadratic Equation puts the headline number first. Quadratic Equation works for one check or many. Quadratic Equation runs in your browser. Math data stays in the tab. Quadratic Equation handles quadratic equation solver with steps the same way as solve quadratic equations by factoring calculator — one of the Math tools on CalcProMaster. Quadratic Equation shows formula, assumptions and result together. Start with Quadratic Equation; the first number takes seconds.
What the defaults give
x² − 5x + 6 = 0 has roots x = 3 and x = 2, with a discriminant of 1.
A discriminant of 1 is a perfect square, so both answers are whole numbers and the expression factors cleanly as (x − 3)(x − 2).
See the roots on a graph
A quadratic ax² + bx + c = 0 has roots where the parabola crosses the x-axis. The discriminant b² − 4ac decides how many: positive gives two real roots, zero a repeated root, negative none.
For x² − 2 = 0 the roots are x = ±√2 ≈ ±1.414 — exactly where the curve meets the axis. Plot it with our free function grapher: type x^2 - 2 and watch where it crosses zero. Related: completing the square and cubic equation solver.
The formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Everything hinges on the part under the root, D = b² − 4ac. Positive means two real roots, zero means one repeated root, negative means the two solutions are complex conjugates. For the default, D = 25 − 24 = 1 and √1 = 1, so x = (5 ± 1)/2 gives 3 and 2.
Worked examples
x² − 5x + 6 → 3.0000 and 2.0000, D = 1
0.5x² − 5x + 6 → 8.6056 and 1.3944, D = 13
2x² − 5x + 6 → complex roots, D = −23
x² − 10x + 6 → 9.3589 and 0.6411, D = 76
x² − 5x + 3 → 4.3028 and 0.6972, D = 13
x² − 5x + 12 → complex roots, D = −23
Halving a from 1 to 0.5 widens the parabola and pushes the roots far apart, because the denominator 2a shrinks while the discriminant grows from 1 to 13. Doubling a to 2 does the opposite and overshoots: the vertex rises above the axis and both solutions become complex.
Reading D before solving
Check D first and you know what kind of answer to expect. In the table above, every complex case has a negative D and every pair of real roots has a positive one. A D of exactly zero would mean the two roots coincide — the curve just touches the axis — and the formula still returns that single value twice.
The ± is not optional. Dropping one sign gives a single root and hides half the solution; the two answers are symmetric about −b/2a, which is the axis of symmetry of the parabola. For the default that axis is x = 2.5, and 3 and 2 sit either side of it.
Roots are not always tidy. An irrational D such as 13 produces the long decimals above, and rounding them too early changes a later check — substituting 8.61 back into 0.5x² − 5x + 6 gives a small residual rather than zero. The tool prints four decimal places so that residual stays small enough to see.
When a is zero the equation is linear, not quadratic, and the formula divides by zero. The page returns a complex-root message only for a genuinely negative discriminant; a zero leading coefficient is not a quadratic at all.
Frequently Asked Questions
What are the roots of x² − 5x + 6?
3 and 2. The discriminant is 25 − 24 = 1, and (5 ± 1)/2 gives both answers directly.
What does the discriminant tell you?
D = b² − 4ac. Positive gives two real roots, zero gives one repeated root, negative gives a pair of complex conjugates.
Why did 2x² − 5x + 6 give no real answer?
Its discriminant is −23. The parabola never touches the x-axis, so the solutions exist only in the complex numbers.
Why are there two answers?
The ± sign produces one root with the square root added and one with it subtracted. Both satisfy the equation.
Can I use this when a is negative?
Yes. The formula holds for any non-zero a; a negative a flips the parabola to open downward, which changes where the vertex sits but not the arithmetic.
Slope-Intercept Form and other Math tools
Slope-Intercept Form · Harmonic Mean · 2×2 Matrix Inverse · Annulus (Ring) Area Calculator · Coefficient of Variation Calculator · Expected Value Calculator · Math Guides