Quadratic Equation Calculator

Solve quadratic equations ax^2 + bx + c = 0. Get solutions, discriminant, vertex, and parabola properties.

Cannot be zero

Quadratic Formula:\nx = (-b +- sqrt(b^2 - 4ac)) / 2a\n\nDiscriminant: Delta = b^2 - 4ac\n- Delta > 0: Two real solutions\n- Delta = 0: One solution (repeated)\n- Delta < 0: Two complex solutions\n\nVertex: (-b/2a, f(-b/2a))\nAxis of Symmetry: x = -b/2a
Example 1:\nx^2 - 5x + 6 = 0\na=1, b=-5, c=6\n\nDiscriminant = (-5)^2 - 4(1)(6) = 25 - 24 = 1\nx = (5 +- sqrt1) / 2 = (5 +- 1) / 2\nx_1 = 3, x_2 = 2\nFactored: (x-3)(x-2) = 0\n\nExample 2:\n2x^2 + 4x + 2 = 0\na=2, b=4, c=2\n\nDiscriminant = 16 - 16 = 0\nx = -4 / 4 = -1 (one solution)\n\nExample 3:\nx^2 + x + 1 = 0\nDiscriminant = 1 - 4 = -3 (negative)\nx = (-1 +- sqrt3i) / 2 (complex)

What is a quadratic equation?

A quadratic equation has the form ax^2 + bx + c = 0, where a != 0. The highest power of x is 2 (squared). Examples: x^2 - 5x + 6 = 0, 2x^2 + 3x - 2 = 0. Graphs as a parabola. Has 0, 1, or 2 real solutions depending on the discriminant.

How do I solve a quadratic equation?

Three main methods: 1) Quadratic formula: x = (-b +- sqrt(b^2 - 4ac)) / 2a (works always). 2) Factoring: (x - p)(x - q) = 0 (if factorable). 3) Completing the square. The quadratic formula is most reliable and works for all quadratic equations.

What is the discriminant and what does it tell us?

Discriminant = b^2 - 4ac. Determines number and type of solutions: If > 0: two different real solutions. If = 0: one repeated real solution (touches x-axis once). If < 0: two complex (imaginary) solutions, no real solutions (doesn't cross x-axis).

What are the vertex and axis of symmetry?

Vertex is the parabola's highest or lowest point. Axis of symmetry is the vertical line through the vertex. Formulas: Axis of symmetry: x = -b/2a. Vertex: (-b/2a, f(-b/2a)). If a > 0, parabola opens up (minimum). If a < 0, opens down (maximum).

What are real-world uses of quadratic equations?

Physics: projectile motion, falling objects. Engineering: bridge arches, satellite dishes. Business: profit maximization, break-even analysis. Computer graphics: curves and animation. Sports: ball trajectories. Economics: supply/demand optimization. Any scenario with squared relationships.