Quadratic Formula Calculator
Solve equations in the form ax² + bx + c = 0. Enter the three coefficients to calculate the discriminant, identify the type of roots, find both solutions and locate the parabola's vertex.
How to use the Quadratic Formula Calculator
- Enter the coefficients a, b and c from ax² + bx + c = 0.
- The discriminant is calculated as b² − 4ac.
- A positive discriminant produces two distinct real roots.
- A zero discriminant produces one repeated real root.
- A negative discriminant produces a complex-conjugate pair. The calculator also finds the vertex coordinates.
Examples
x² − 3x + 2 = 0
With a = 1, b = −3 and c = 2, the discriminant is 1 and the roots are x = 1 and x = 2.
x² + 2x + 1 = 0
The discriminant is zero, so the equation has one repeated real root at x = −1.
x² + 1 = 0
The discriminant is −4, so there are no real roots. The complex roots are i and −i.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a not equal to zero, the roots are found from negative b plus or minus the square root of b² − 4ac, all divided by 2a.
What is the discriminant?
The discriminant is b² − 4ac. Its sign determines whether a quadratic has two real roots, one repeated real root or two complex roots.
What happens if a equals zero?
The equation is no longer quadratic. When b is nonzero, it becomes a linear equation and can be solved as x = −c/b.
Can this calculator show complex roots?
Yes. When the discriminant is negative, the calculator displays the real and imaginary parts of both complex roots.
How is the vertex of a quadratic found?
The vertex x-coordinate is −b divided by 2a. Substituting that x-value into ax² + bx + c gives the vertex y-coordinate.
More math tools
Graphing Calculator
PopularGraph functions, trace points, view tables and analyze equations.
Open tool →Scientific Calculator
PopularAdvanced calculator for trig, logs, roots, powers, factorials and more.
Open tool →Slope Calculator
PopularCalculate slope from two points plus line equation, angle, grade and point distance.
Open tool →Factor Calculator
PopularFind factors, factor pairs and prime factorization for a whole number.
Open tool →Exponent Calculator
PopularCalculate powers using positive, negative or decimal exponents.
Open tool →Percentage Calculator
PopularCalculate percentages, increases and decreases.
Open tool →