Algebra
Quadratic Formula Calculator — Solve for x
Enter the coefficients a, b, and c of any quadratic equation to instantly find both roots, the discriminant, vertex, and axis of symmetry.
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📝 Enter Coefficients (ax²+bx+c=0)
📊 Result
Enter a, b and c to solve the equation
Disclaimer: This tool performs standard mathematical calculations for educational and reference purposes. Always double-check results independently for graded coursework, exams, or safety-critical applications.
The Quadratic Formula
Any equation of the form ax² + bx + c = 0 (where a ≠ 0) can be solved with the quadratic formula: x = (−b ± √(b²−4ac)) / 2a. The value under the square root, b²−4ac, is called the discriminant, and its sign tells you what kind of roots the equation has before you even finish solving.
Reading the Discriminant
- Discriminant > 0: two distinct real roots — the parabola crosses the x-axis at two points
- Discriminant = 0: one repeated real root — the parabola touches the x-axis at exactly one point (its vertex)
- Discriminant < 0: two complex (imaginary) roots — the parabola never touches the x-axis
Worked Example
Solve x² − 5x + 6 = 0 (a=1, b=−5, c=6).
- Step 1: Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1
- Step 2: Since discriminant > 0, there are two real roots
- Step 3: x = (5 ± √1) / 2 = (5 ± 1) / 2, giving x = 3 and x = 2
Frequently Asked Questions
The equation is no longer quadratic (it becomes linear, bx+c=0), so this calculator requires a non-zero value for a. If you need to solve a linear equation, isolate x directly: x = −c/b.
Complex roots appear when the discriminant is negative, meaning the parabola never crosses the x-axis. They're written in the form p ± qi, where i is the imaginary unit (√−1). These are valid mathematical solutions, just not real-number x-intercepts.
The roots are where the parabola crosses the x-axis (y=0). The vertex is the parabola's highest or lowest point, always located at x = −b/2a — it exists regardless of whether the parabola crosses the x-axis at all.
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