Quadratic Equation Solver
Solve quadratic equations (ax² + bx + c = 0) instantly. Find roots, discriminant, and vertex coordinates.
Embed this toolEnter Coefficients
1x² + -5x + 6 = 0
Results
Root x₁
3
Root x₂
2
Discriminant (b² - 4ac)
1.0000
Vertex
(2.5000, -0.2500)
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Understanding the Quadratic Formula
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. The solutions to this equation are given by the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
The expression under the square root, b² - 4ac, is called the discriminant. It determines the nature of the roots:
- Positive discriminant: Two distinct real roots.
- Zero discriminant: Exactly one real root (a repeated root).
- Negative discriminant: No real roots; two complex roots.
The vertex of the parabola y = ax² + bx + c is located at (-b/2a, f(-b/2a)). It represents the highest or lowest point on the graph, depending on whether the parabola opens downward or upward.
Frequently Asked Questions
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It gives the solutions (roots) to any quadratic equation in the form ax² + bx + c = 0, where a ≠ 0.