Quadratic Formula Calculator
Solve ax² + bx + c = 0
Coefficient of x²
Coefficient of x
Constant
Try an example
Step-by-Step Solution
Step 1 — Identify the coefficients
a = 1, b = -3, c = 2
Step 2 — Calculate the discriminant
Δ = b² − 4ac
Δ = (-3)² − 4(1)(2)
Δ = 9 − 8
Δ = 1
Step 3 — Apply the quadratic formula
x = (−b ± √Δ) / 2a
Step 4 — Calculate both roots
x₁ = 2
x₂ = 1
Step 5 — Verify the solutions
Substitute each root back into the original equation to confirm it equals 0.
Factored Form
(x − 2)(x − 1) = 0
Solutions
x = 2
x = 1
Graph
Mathematical Insight
x = −b / 2a
Vertex: (1.5, -0.25)
Axis of symmetry: x = 1.5
Two real solutions
The parabola crosses the x-axis twice.
This quadratic formula calculator (or quadratic equation calculator) solves any equation in the form ax^2 + bx + c = 0 for x, showing both roots and whether they are real or complex. It handles solving quadratic equations calculator problems for any coefficients a, b, and c, including cases with a negative discriminant.
Enter the coefficients a, b, and c, and the calculator returns both roots using the quadratic formula.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| x² − 3x + 2 = 0 (a=1, b=−3, c=2) | x = 2 or x = 1 | Discriminant = (−3)² − 4(1)(2) = 9 − 8 = 1. x = [3 ± √1] ÷ 2 = (3±1)/2, giving x = 2 or x = 1. |
| x² + 2x + 5 = 0 (a=1, b=2, c=5) | x = −1 + 2i or x = −1 − 2i | Discriminant = 2² − 4(1)(5) = 4 − 20 = −16, which is negative. √−16 = 4i. x = [−2 ± 4i] ÷ 2 = −1 ± 2i, a pair of complex roots. |
The Quadratic Formula Explained
The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / (2a). The expression under the square root, b^2 - 4ac, is called the discriminant - it determines whether the equation has two real roots (positive discriminant), one repeated real root (discriminant of zero), or two complex roots (negative discriminant). The formula's derivation comes from a technique called completing the square, applied to the general equation ax² + bx + c = 0; versions of this method were known to Babylonian mathematicians nearly 4,000 years ago, though the fully general symbolic formula in its modern form wasn't published until the 16th and 17th centuries as algebraic notation matured.
Reading the Discriminant
A positive discriminant means the parabola crosses the x-axis at two points (two real solutions). A discriminant of exactly zero means the parabola just touches the x-axis at one point (one repeated solution). A negative discriminant means the parabola never crosses the x-axis, so the solutions involve imaginary numbers.
How to Use the Quadratic Formula Calculator
- Enter the coefficient of x².
- Enter the coefficient of x.
- Enter the constant.
- Review the equation preview.
- Check the discriminant and roots.
- Follow the step-by-step solution.
Frequently Asked Questions
What is the quadratic formula?
x = (-b +/- sqrt(b^2 - 4ac)) / (2a), used to solve any equation in the form ax^2 + bx + c = 0 for its roots.
How do I know if a quadratic equation has real or complex roots?
Check the discriminant, b^2 - 4ac: positive means two real roots, zero means one repeated real root, and negative means two complex roots.
How do I solve a quadratic equation step by step?
Identify a, b, and c from the equation, calculate the discriminant b^2 - 4ac, then plug all three values into the quadratic formula to find both roots.
How do I use this quadratic formula calculator?
Enter the known values, review the units or settings, and the calculator updates the result instantly. The formula and example on this page show how the answer is produced.
What does the Quadratic Formula Calculator calculate?
Solve ax^2 + bx + c = 0 quadratic equations. It is designed for fast browser-based calculations without sign-up, downloads, or manual spreadsheet setup.
What formula does this quadratic formula use?
The formula is: x = [−b ± √(b² − 4ac)] ÷ (2a). a, b, and c are the coefficients of ax² + bx + c = 0 (a ≠ 0). The ± symbol means the formula produces two roots: one using +√(discriminant) and one using −√(discriminant). When the discriminant (b² − 4ac) is negative, its square root is an imaginary number, which is why negative-discriminant equations produce two complex roots instead of real ones.
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Last updated: September 29, 2026.