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- Solves ax² + bx + c = 0 for x using the quadratic formula.
- Shows the discriminant and whether roots are real, repeated or complex.
- Gives the vertex, axis of symmetry, y-intercept and factored form.
- Works with decimals, negative coefficients and complex roots.
How to use the quadratic formula calculator
Write your equation in standard form
Rearrange it to ax² + bx + c = 0.
Enter a, b and c
Use 0 for a missing term. For x² − 9 = 0, enter a = 1, b = 0, c = −9.
Read the roots
The calculator shows both solutions, or one repeated root, or the complex pair.
Check the working
The steps show the discriminant, its square root and the final division.
How the quadratic formula works
The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Discriminant D = b² − 4ac
D > 0 gives two real roots, D = 0 gives one repeated root, and D < 0 gives two complex roots.
Vertex of the parabola
x = −b ÷ 2a
y = c − b² ÷ 4a
The parabola opens upward when a > 0 and downward when a < 0.
Example 1: x² − 3x + 2 = 0
a = 1, b = −3, c = 2.
- Discriminant 9 − 8 = 1
- Roots (3 ± 1) ÷ 2 = 2 and 1
- Factored form (x − 2)(x − 1)
- Vertex (1.5, −0.25)
Two different real roots, so the parabola crosses the x-axis twice.
Example 2: 2x² + 3x − 2 = 0
a = 2, b = 3, c = −2.
- Discriminant 9 + 16 = 25
- Roots (−3 ± 5) ÷ 4 = 0.5 and −2
- Factored form 2(x − 0.5)(x + 2)
When a is not 1, it stays in front of the factored form.
Example 3: complex roots
x² + 2x + 5 = 0.
- Discriminant 4 − 20 = −16
- Roots −1 ± 2i
- Vertex (−1, 4)
A repeated root happens when D = 0, for example x² − 4x + 4 = 0 has the single root x = 2.
Where a quadratic formula calculator helps
Algebra and exams
Check quadratic equation answers and learn the method from the steps.
Physics problems
Solve projectile motion and other equations that include a squared term.
Area and geometry
Find a side length when an area problem leads to a quadratic equation.
Business break-even
Solve profit models where revenue or cost has a squared term.
Find the slope of a straight line with the slope calculator, simplify the square root of the discriminant with the square root calculator, and raise numbers to any power with the exponent calculator.
Quadratic formula calculator FAQ
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any equation of the form ax² + bx + c = 0 where a is not 0.
What does the discriminant tell me?
It tells you the type of roots. Positive means two real roots, zero means one repeated root, and negative means two complex roots.
What if a = 0?
Then the equation is linear, not quadratic. The calculator solves it as bx + c = 0, giving x = −c ÷ b.
What are complex roots?
When the discriminant is negative, the square root involves i, the imaginary unit where i² = −1. The roots come as a pair like −1 ± 2i.
How do I enter a missing term?
Use 0. For example 3x² − 12 = 0 has b = 0.
How do I check my answer?
Put each root back into ax² + bx + c. The result should be 0. You can also check that the roots add to −b/a and multiply to c/a.