Quadratic Formula Calculator
Solve quadratic equations using the quadratic formula with our easy-to-use calculator. Enter the coefficients a, b, and c of the quadratic equation ax² + bx + c = 0 to find the roots, discriminant, vertex, and graph of the parabola.
The Quadratic Formula
This formula gives the solutions to any quadratic equation of the form:
Results
Equation:
Solutions:
x = 1
Discriminant:
Vertex:
Root Nature:
About the Quadratic Formula
The quadratic formula is a fundamental tool in algebra that provides the solution(s) to any quadratic equation. A quadratic equation is a second-order polynomial equation in a single variable x, with a coefficient a that is not equal to zero.
Understanding the Components
The quadratic formula calculates the roots of the equation ax² + bx + c = 0, where:
- a is the coefficient of the x² term (must be non-zero)
- b is the coefficient of the x term
- c is the constant term
The Discriminant
The expression under the square root in the quadratic formula, b² - 4ac, is called the discriminant (Δ). It determines the nature of the roots:
Positive Discriminant (Δ > 0)
Two distinct real roots. The parabola intersects the x-axis at two different points.
Zero Discriminant (Δ = 0)
One real root (a repeated root). The parabola touches the x-axis at exactly one point (the vertex).
Negative Discriminant (Δ < 0)
No real roots (two complex conjugate roots). The parabola does not intersect the x-axis.
Vertex of the Parabola
The vertex of the quadratic function f(x) = ax² + bx + c is the point where the parabola changes direction. The x-coordinate of the vertex is given by -b/2a, and the y-coordinate can be found by substituting this x-value back into the equation.
Graphical Interpretation
The graph of a quadratic equation is a parabola that opens upward if a > 0 and downward if a < 0. The roots of the equation correspond to the x-intercepts of the parabola.