Calculator Tools
Quadratic Equation Solver
Solve a x^2 + b x + c = 0 for real or complex roots. Discriminant, vertex, axis of symmetry, factored form, Vieta's sums, and step-by-step working.
Quadratic equation solver
Enter the coefficients of a x^2 + b x + c = 0
Each field accepts integers, decimals, or simple fractions like 3/4. Use a leading minus for negative values.
Equation
x^2 - 5x + 6 = 0
Discriminant
1
Axis of symmetry
x = 2.5
y-intercept
(0, 6)
Roots
Two distinct real roots. The parabola crosses the x-axis at two points.
x1
3
x2
2
Exact form
- x1 = (-(-5) + 1) / (2 * 1)
- x2 = (-(-5) - 1) / (2 * 1)
Step-by-step
Each line shows the substitution or simplification used to reach the next result.
- Start with x^2 - 5x + 6 = 0.
- Identify the coefficients: a = 1, b = -5, c = 6.
- Compute the discriminant D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1.
- D > 0, so the equation has two distinct real roots x = (-b +/- sqrt(D)) / (2a).
- sqrt(1) = 1, so the radical disappears: x = (-(-5) +/- 1) / (2 * 1).
- Plug in to get x1 = 3 and x2 = 2.
- The vertex of the parabola y = x^2 - 5x + 6 is (2.5, -0.25).
- The axis of symmetry is x = -b / (2a) = 2.5.
- By Vieta's formulas, sum of roots = -b/a = 5 and product of roots = c/a = 6.
The quadratic formula
x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
- Works for every quadratic with a non-zero leading coefficient, even when no rational factor exists.
- The expression under the square root, b^2 - 4ac, is the discriminant. Its sign decides whether the roots are real or complex.
- When the discriminant is a perfect square the roots are rational; otherwise they involve an irrational surd.
Reading the discriminant
- D > 0 means two distinct real roots. The parabola crosses the x-axis twice.
- D = 0 means one repeated real root. The parabola touches the x-axis at one point.
- D < 0 means two complex conjugate roots. The parabola never touches the x-axis.
- When D is a perfect square, the equation factors over the rationals, and a clean factored form is shown above.
How to use
- Enter the three coefficients a, b, and c from a x^2 + b x + c = 0. Each field accepts integers, decimals, fractions like 3/4, and leading minus signs.
- Read the headline equation and the discriminant panel. The sign of the discriminant tells you whether the roots are two real, one repeated real, or two complex conjugate values.
- Check the Roots section for decimal and exact symbolic answers. The exact form simplifies square roots to k*sqrt(m) when possible.
- Use the side panel for the vertex, axis of symmetry, y-intercept, sum and product of roots from Vieta's formulas, and the factored form when both roots are simple rationals.
- Open the Step-by-step block to see the derivation, or click Copy full solution to copy the entire result as plain text.
About this tool
Quadratic Equation Solver answers every part of the standard quadratic question in one pass. Enter the coefficients a, b, and c from a x^2 + b x + c = 0 and the tool computes the discriminant b^2 - 4ac, classifies the roots, and prints them in both decimal and exact symbolic form. When the discriminant is positive the result is two distinct real roots; when it is zero the result is one repeated real root; when it is negative the result is two complex conjugate roots written as a real part plus a simplified imaginary surd. The exact form simplifies sqrt(D) to k*sqrt(m) with m square-free whenever possible, so 2x^2 - 4x - 6 = 0 is reported as x = 1 +/- 2 and 3x^2 - 7x + 2 = 0 is reported with its rational factors instead of a long decimal. The vertex (-b/2a, c - b^2/4a), the axis of symmetry, the y-intercept (0, c), the sum of roots -b/a, and the product of roots c/a all appear in the result panel. When both roots are simple rationals, the tool prints the factored form a(x - r1)(x - r2) so you can see the equation split into linear factors. A step-by-step block mirrors the textbook derivation: identify the coefficients, evaluate the discriminant, apply the quadratic formula, simplify the surd, and write the final roots. Everything runs locally in the browser, so the equations you experiment with never leave your device.
Free to use. Works in your browser. No signup, no login.
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