Quadratic Formula Calculator
The roots of ax² + bx + c = 0, real or complex, the discriminant and the vertex, computed in a form that stays accurate when one root is tiny.
Result
First root: 2
Roots shown to ten significant figures.
- Second root
- 1
- Discriminant, b² − 4ac
- 1
- The roots are
- Two distinct real roots.
- Vertex x, the axis of symmetry
- 1.5
- Vertex y
- -0.25
https://calcgallery.com/algebra-numbers/quadratic-formula-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Quadratic Formula Calculator
- Write your equation in the form ax² + bx + c = 0.
- Enter a, b and c, including any minus signs.
- Read the two roots, whether they are real or complex, and the vertex of the parabola.
The formula
x = (−b ± √(b² − 4ac)) ÷ 2aThe discriminant, D = b² − 4ac, decides what kind of roots there are. When it is negative, its square root is imaginary, and the roots are −b ÷ 2a ± i√(−D) ÷ 2a.
The vertex
The parabola y = ax² + bx + c turns at x = −b ÷ 2a, its axis of symmetry, where y = c − b² ÷ 4a. It opens upward when a is positive and downward when a is negative.
A stable form
The page computes q = −(b + sign(b)√D) ÷ 2 and takes the roots as q ÷ a and c ÷ q. The two expressions give the same roots as the usual formula, but they never subtract two nearly equal numbers, so neither root loses its digits.
With your numbers
- D = b² − 4ac = 1
- x = (−b ± √D) ÷ 2a, in the stable form
Quick reference
Roots shown to ten significant figures.
| Example | First root |
|---|---|
| x² − 3x + 2 | 2 |
| x² + 2x + 1 | −1 |
| x² + 1 | 0 + 1i |
| x² − 2x − 3 | 3 |
| x² + 100,000,000x + 1 | −1 × 10^-8 |
| 2x + 4, with a = 0 | −2 |
Common questions about the Quadratic Formula Calculator
- What does the discriminant tell me?
- The number of real roots. If b² − 4ac is positive there are two different real roots, where the parabola crosses the x-axis twice. If it is zero there is one repeated root, where it just touches. If it is negative there are no real roots; the two roots are complex, a pair of the form p + qi and p − qi.
- How do I read a complex root?
- As a real part and an imaginary part: 0 + 1i means zero plus one times i, the square root of −1. For x² + 1 = 0 the roots are i and −i, because i² = −1. Complex roots of a quadratic with real coefficients always come in such pairs.
- Why does the calculator not use the formula exactly as taught?
- Because on a computer the textbook formula can lose one root completely. For x² + 100,000,000x + 1 = 0 it subtracts two nearly equal numbers and gets −7.45 × 10⁻⁹ for a root that is really −1 × 10⁻⁸. The page uses an equivalent form that avoids the subtraction, so both roots are accurate.
- What if a is zero?
- Then there is no x² term and the equation is linear, bx + c = 0, with the single root −c ÷ b. The page solves it and says so.