Quadratic Formula Calculator
Solve quadratic equations instantly. Find roots, discriminant, and vertex with visual graph.
Formula
x = (-b ± √(b² - 4ac)) / 2a
Finds the x-intercepts of a quadratic function.
Worked Examples
Example 1: Standard Case
Problem:x² - 5x + 6 = 0
Solution:a=1, b=-5, c=6. Discriminant = 25 - 24 = 1. Roots = (5 ± 1)/2.
Result:x = 3, x = 2
Example 2: Complex Roots
Problem:x² + x + 1 = 0
Solution:a=1, b=1, c=1. Discriminant = 1 - 4 = -3.
Result:x = -0.5 ± 0.866i
Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a. It calculates the roots (solutions) of a quadratic equation ax² + bx + c = 0.
What is the discriminant?
The discriminant is Δ = b² - 4ac. It determines the nature of the roots: positive means 2 real roots, zero means 1 real root, negative means 2 complex roots.
What if the discriminant is negative?
If b² - 4ac < 0, the equation has no real solutions. The graph (parabola) does not touch the x-axis. The solutions involve imaginary numbers (i).
What if the discriminant is zero?
If b² - 4ac = 0, there is exactly one real solution (a repeated root). The vertex of the parabola sits on the x-axis.
Can I solve quadratics without this formula?
Yes. Methods include Factoring (if integers work), Completing the Square, or Graphing. The formula is reliable because it works for ALL quadratics.
How do I find the Vertex?
The x-coordinate of the vertex is -b / 2a. Plug this x back into the equation to find the y-coordinate.
What is the Standard Form of a quadratic?
y = ax² + bx + c. The graph is a parabola opening up (if a > 0) or down (if a < 0).
Where is the y-intercept?
The y-intercept is simply "c" (the constant term), because when x=0, y=c.
How are quadratics used in real life?
They model projectile motion (throwing a ball), satellite dish shapes, maximizing area/profit, and bridge arches.
Why must the coefficient a be non-zero?
The quadratic formula divides by 2a, so a = 0 makes it undefined. More fundamentally, with a = 0 the x² term disappears and ax² + bx + c = 0 reduces to the linear equation bx + c = 0, whose single solution is x = -c/b. There is no parabola, no vertex and no discriminant to interpret, which is why Quadratic Formula Calculator flags a = 0 as not quadratic.
Background & Theory
Quadratic Equation:
An equation of the form ax² + bx + c = 0, where a ≠ 0.
Quadratic Formula:
x = (-b ± √(b² - 4ac)) / 2a
Discriminant (Δ):
Δ = b² - 4ac
- Δ > 0: Two distinct real roots.
- Δ = 0: One real root (repeated).
- Δ < 0: Two complex conjugate roots.
Vertex:
The turning point of the parabola.
x = -b / 2a