Quadratic Equation Solver
Solve equations in the form: ax² + bx + c = 0
How to Use This Quadratic Equation Solver
Enter your coefficients
Input the values for a, b, and c from your equation ax² + bx + c = 0. Make sure a is not zero.
Click Solve
The calculator will compute the discriminant and find the roots using the quadratic formula.
Review the results
You will see both roots (x₁ and x₂), the discriminant value, the nature of roots, and the vertex of the parabola.
Understanding Quadratic Equations
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic equation is a parabola.
The quadratic formula gives you the roots (solutions) of any quadratic equation. The ± symbol means there are typically two solutions: one using addition and one using subtraction.
The Discriminant and Nature of Roots
| Discriminant (Δ) | Nature of Roots | Graph Behavior |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | One repeated real root | Parabola touches x-axis once |
| Δ < 0 | Two complex conjugate roots | Parabola does not cross x-axis |
The discriminant (Δ = b² - 4ac) tells you what kind of solutions to expect before solving.
Finding the Vertex
The vertex is the highest or lowest point on a parabola. For a quadratic equation, the vertex coordinates (h, k) are:
If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, the parabola opens downward and the vertex is a maximum.
Common Quadratic Equation Examples
x² - 5x + 6 = 0
a=1, b=-5, c=6 → Roots: x=2, x=3
2x² + 4x - 6 = 0
a=2, b=4, c=-6 → Roots: x=1, x=-3
x² - 4x + 4 = 0
a=1, b=-4, c=4 → One root: x=2 (repeated)
x² + x + 1 = 0
a=1, b=1, c=1 → Complex roots: -0.5 ± 0.866i
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves any quadratic equation ax² + bx + c = 0. The formula comes from completing the square on the general quadratic equation.
What does the discriminant tell me?
The discriminant (b² - 4ac) reveals the nature of the roots. Positive means two real roots, zero means one repeated root, and negative means two complex roots. You can check this before solving to know what to expect.
Can a be zero in a quadratic equation?
No. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula would also fail because it divides by 2a. Make sure your equation has an x² term.
What is the vertex of a parabola?
The vertex is the turning point of the parabola — its highest or lowest point. The x-coordinate is -b/2a, and you find the y-coordinate by plugging this back into the equation. The vertex tells you the maximum or minimum value of the quadratic function.
When do quadratic equations have complex roots?
Complex roots occur when the discriminant is negative (b² - 4ac < 0). This happens when the parabola does not cross the x-axis. The roots come in conjugate pairs: a + bi and a - bi, where i is the imaginary unit (√-1).
Other Free Tools
Linear Equation Solver
Linear Equation Solver – Solve ax + b = 0
Determinant Calculator
Determinant Calculator
Matrix Addition Calculator
Matrix Addition Calculator
Matrix Multiplication Calculator
Matrix Multiplication Calculator
Inverse Matrix Calculator
Inverse Matrix Calculator
Expression Evaluator
Expression Evaluator