Quadratic Equation Solver
Quadratic equations have form ax² + bx + c = 0. Solve using quadratic formula: x = [-b ± √(b²-4ac)] / 2a. Discriminant (b²-4ac) determines solutions: positive = 2 real roots, zero = 1 real root (double), negative = 2 complex roots. For example, x² - 5x + 6 = 0: a=1, b=-5, c=6. x = [5 ± √(25-24)] / 2 = [5 ± 1] / 2. Solutions: x=3 and x=2. Alternative: factoring (x-2)(x-3)=0 or completing the square. Use in physics (projectile motion), engineering, and optimization.
Solve quadratic equations (ax² + bx + c = 0) using the quadratic formula. Find roots, vertex, discriminant, and axis of symmetry.
Solving
x² - 5x + 6 = 0
ax² + bx + c = 0
a (coefficient of x²)
b (coefficient of x)
c (constant)
Solutions
Two distinct real roots
Discriminant (b² - 4ac)
1
> 0: Two real roots
Vertex
(2.5, -0.25)
Minimum point
Axis of Symmetry
x = 2.5
Y-Intercept
(0, 6)
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
- Discriminant > 0: Two distinct real roots
- Discriminant = 0: One repeated (double) root
- Discriminant < 0: Two complex conjugate roots
- Vertex form: y = a(x - h)² + k where (h, k) is the vertex
How to Use
- Enter your value in the input field
- Click the Calculate/Convert button
- Copy the result to your clipboard
Frequently Asked Questions
- What is a quadratic equation?
- A quadratic equation has the form ax² + bx + c = 0 where a≠0. The graph is a parabola. Examples: x² - 5x + 6 = 0, 2x² + 3x - 5 = 0. Has at most 2 solutions (roots). Used in physics, engineering, economics, and many sciences.
- What is the quadratic formula?
- x = (-b ± √(b² - 4ac)) / 2a. This formula solves any quadratic equation. The ± means two solutions: one with + and one with -. Example: x² - 5x + 6 = 0 gives x = (5 ± √1) / 2 = 3 or 2.
- What is the discriminant?
- The discriminant is b² - 4ac, the part under the square root. If discriminant > 0: two distinct real roots. If = 0: one repeated root. If < 0: two complex conjugate roots (no real solutions). Discriminant reveals solution nature without solving.
- What is the vertex of a parabola?
- The vertex is the highest or lowest point of a parabola. Located at x = -b/(2a). For y = ax² + bx + c, vertex is at (-b/2a, f(-b/2a)). If a > 0, vertex is minimum (parabola opens up). If a < 0, vertex is maximum.
- How do I factor a quadratic?
- For x² + bx + c, find two numbers that multiply to c and add to b. x² - 5x + 6: numbers are -2 and -3 (multiply to 6, add to -5). So x² - 5x + 6 = (x - 2)(x - 3). Set each factor to 0: x = 2 or x = 3.