Quadratic Equation Calculator
Solve quadratic equations of the form ax² + bx + c = 0
ax² + bx + c = 0
About Quadratic Equations
A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a ≠ 0. The solutions are called roots and can be found using the quadratic formula.
Quadratic Formula
x = (-b ± √(b² - 4ac)) / (2a)
Where the discriminant Δ = b² - 4ac determines the nature of the roots:
- Δ > 0: Two distinct real roots
- Δ = 0: One repeated real root
- Δ < 0: Two complex conjugate roots
Key Properties
- Vertex: The turning point of the parabola at (-b/2a, f(-b/2a))
- Axis of Symmetry: The vertical line x = -b/2a
- Y-intercept: The point where the parabola crosses the y-axis at (0, c)
- Direction: Opens upward if a > 0, downward if a < 0
Examples
Example 1
x² - 5x + 6 = 0
a = 1, b = -5, c = 6
Roots: x = 2, x = 3
Example 2
x² - 4x + 4 = 0
a = 1, b = -4, c = 4
Root: x = 2 (repeated)
Example 3
x² + x + 1 = 0
a = 1, b = 1, c = 1
Complex roots