Quadratic Formula Calculator
Solve algebraic equations, quadratic formulas, systems of equations, and polynomial problems.
Quadratic Equation
Coefficient of x² (cannot be 0)
Coefficient of x
Constant term
📐 Common Examples
Quadratic Formula
Solutions
Solution 1
Solution 2
Discriminant
Two real
Vertex
Minimum
Axis of Symmetry
Y-Intercept
When x = 0
Factored Form
Step-by-Step Solution
Parabola Properties
How it works
The quadratic formula solves any equation of the form ax² + bx + c = 0. It returns the values of x where the parabola crosses zero. The piece under the square root — the discriminant — tells you whether there are two, one, or no real solutions.
Quadratic formula
x = [ −b ± √(b² − 4ac) ] ÷ 2a
- a, b, c
- coefficients from ax² + bx + c = 0
- b² − 4ac
- the discriminant (sign decides the number of roots)
Worked example
- Solve x² − 5x + 6 = 0 (a=1, b=−5, c=6)
- Discriminant = (−5)² − 4·1·6 = 25 − 24 = 1
- x = (5 ± √1) ÷ 2
x = 3 or x = 2.
Good to know
- Discriminant > 0 → two real roots; = 0 → one repeated root; < 0 → two complex roots.
- If the quadratic factors easily (like x² − 5x + 6 = (x−2)(x−3)), factoring is faster than the formula.
- The roots sum to −b/a and multiply to c/a — a quick sanity check.
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Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a solves any quadratic equation of the form ax² + bx + c = 0. It always works, even when the equation can't be factored neatly.
What does the discriminant tell me?
The discriminant is b² - 4ac. If it's positive there are two distinct real roots, if zero there's exactly one repeated real root, and if negative the two roots are complex conjugates.
When should I use the quadratic formula instead of factoring?
Factor when integer roots are obvious; otherwise use the formula. It's the reliable fallback for messy coefficients, irrational roots, or any time factoring isn't quick.
How do I get an equation into standard form first?
Move every term to one side so the equation reads ax² + bx + c = 0, then identify a, b, and c including their signs. For example, 2x² = 5x - 3 becomes 2x² - 5x + 3 = 0, so a=2, b=-5, c=3.
What happens if a = 0?
The equation is no longer quadratic — it's linear (bx + c = 0), and the quadratic formula breaks down because of division by 2a. Solve it directly as x = -c/b.