Quadratic formula calculator
Solve ax^2 + bx + c = 0 and see the discriminant, axis of symmetry, and both roots.
What this calculator covers
Use this quadratic formula calculator when you have an equation in standard form and want the roots without hiding the discriminant.
The page keeps the real-versus-complex split explicit, so you can see whether the parabola crosses the x-axis or only has complex conjugate solutions.
Frequently asked questions
- What does the discriminant tell you about the roots?
- The discriminant is b² − 4ac. A positive discriminant means two distinct real roots, zero means one repeated real root, and a negative discriminant means two complex conjugate roots with no real x-intercepts. Checking the discriminant first tells you what type of solution to expect before solving.
- What are complex roots and when do they appear?
- Complex roots contain an imaginary component (written with i, where i = √−1) and appear whenever the discriminant is negative. They come in conjugate pairs, meaning if one root is a + bi the other is a − bi. The parabola does not cross the x-axis in this case.
- Why must the coefficient a be non-zero?
- If a equals zero, the x² term disappears and the equation becomes linear rather than quadratic. A linear equation has at most one solution and is solved differently, so the quadratic formula does not apply.
- What is the axis of symmetry?
- The axis of symmetry is the vertical line x = −b / (2a) that passes through the vertex of the parabola. It lies exactly halfway between the two roots when they are real, and equals the real part of the complex roots when the discriminant is negative.
Tool
Run the calculation
Result
RESULT · ROOTS
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Primary result
x1 = 5, x2 = -2
For 1x^2 + -3x + -10 = 0, the discriminant is 49 and the roots are 5 and -2.
- Discriminant
- 49
- Axis of symmetry
- x = 1.5
- Root 1
- 5
- Root 2
- -2
Step-by-step solution
- 1.Compute the discriminant b^2 - 4ac = -3^2 - 4(1)(-10) = 49.
- 2.Take the square-root term from the discriminant.
- 3.Plug the values into x = (-b +/- sqrt(discriminant)) / 2a to get x1 = 5 and x2 = -2.
Walkthrough
Visual walkthrough
The quadratic formula breaks into three parts: the discriminant, the square-root term, and the final division by 2a.
01
Measure the discriminant
b^2 - 4ac = 49
The discriminant decides whether the parabola crosses the x-axis twice, once, or only in the complex plane.
02
Build the square-root term
sqrt(49)
A non-negative discriminant gives a real square root, while a negative one turns the square root into an imaginary term.
03
Solve both roots
Using the plus and minus branches captures both x-intercepts or both complex conjugates of the quadratic.
5; -2