Quadratic Equation
ax² + bx + c = 0
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Quadratic Formula Calculator Guide
Solve any quadratic equation of the form ax² + bx + c = 0. Enter the coefficients a, b, and c to find the roots, the discriminant, and the vertex of the parabola.
x = [−b ± √(b² − 4ac)] ÷ 2a. The part under the root, b² − 4ac, is the discriminant, which tells you how many real solutions exist.
How Do I Solve a Quadratic Equation?
Plug a, b, and c into the quadratic formula. For example, x² − 5x + 6 = 0 has a discriminant of 1 and two roots, x = 3 and x = 2. When the discriminant is negative, the roots are complex numbers.
What the Discriminant Tells You
If the discriminant is positive, there are two real roots; if it is zero, there is one repeated real root; if it is negative, there are two complex roots. The vertex, at x = −b ÷ 2a, is the highest or lowest point of the parabola.
Related: Square Root, Exponent Calculator, and Scientific Notation.
Frequently Asked Questions
What is the quadratic formula?
x = [−b ± √(b² − 4ac)] ÷ 2a, used to solve any equation of the form ax² + bx + c = 0.
What is the discriminant?
The expression b² − 4ac; it is positive for two real roots, zero for one, and negative for complex roots.
Can a quadratic have no real solutions?
Yes — when the discriminant is negative, both roots are complex numbers rather than real.
What is the vertex of a parabola?
The turning point, located at x = −b ÷ 2a, which is the minimum or maximum of the quadratic.
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