Maths / Algebra
Quadratic Equation Solver
Solve ax² + bx + c = 0 with the working shown: the discriminant, exact roots as simplified square roots, decimal values, complex roots, the vertex, and a graph.
Quadratic Equation Solver: The solver uses the quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac decides the kind of roots: two real roots when it is positive, one repeated root when it is zero, and two complex roots when it is negative. With whole-number coefficients, roots are also given exactly, with the square root simplified, such as 1 ± √2. The decimal roots are computed in a way that avoids losing precision when b is much larger than a and c. Runs 100% locally in your browser with zero server file uploads.
- Category
- School & study tools
- Runs
- In your browser
- Cost
- Free · no sign-up
- Availability
- Ready to use
Runs entirely in your browser
x² − 3x − 4 = 0
Discriminant: Δ = b² − 4ac = 9 + 16 = 25
x = (−b ± √Δ) ÷ 2a
x₁ = -1
x₂ = 4
Δ > 0: two real roots, where the parabola crosses the x-axis.
Vertex: (1.5, -6.25) · opens upwards
Other ways to solve
Factoring is quickest when the roots are whole numbers: x² − 5x + 6 = (x − 2)(x − 3), so x = 2 or 3. Completing the square rewrites ax² + bx + c as a(x − h)² + k, which also gives the vertex (h, k). The formula always works, which is why the solver uses it.
To see the whole curve and where it meets other graphs, use the graphing calculator; to simplify a root by hand, the square root calculator.
Sum and product of the roots
For any quadratic, the two roots add up to −b ÷ a and multiply to c ÷ a. A quick check: for x² − 3x − 4, the roots 4 and −1 add to 3 and multiply to −4.
How to use it
- Enter a, b, and c, for ax² + bx + c = 0; use negative numbers for minus signs.
- Read the discriminant and the roots, exact and as decimals.
- Check the graph: the roots are where the parabola meets the x-axis.
Privacy & limitations
The calculation runs in your browser.
Related tools
Frequently asked questions
What if a is 0?
Then the equation is linear, bx + c = 0, with the single solution x = −c ÷ b; the solver says so.
How do I enter x² − 4x = 0?
Enter a = 1, b = −4, and c = 0; a missing term has a coefficient of 0.
What do complex roots mean?
The parabola never crosses the x-axis, so there is no real solution; the roots are a ± bi, where i is the square root of −1.
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