Quadratic Equation Solver

The discriminant (b² - 4ac) tells you how many real solutions exist without solving the equation. Positive: 2 real solutions. Zero: 1 real solution (the parabola just touches the x-axis). Negative: no real solutions (the parabola doesn't cross the x-axis).

When Δ < 0, the square root of a negative number appears. These are complex (imaginary) numbers of the form a + bi, where i = √(-1). Complex roots come in conjugate pairs. They mean the parabola doesn't intersect the x-axis on the real number plane.

Yes — by factoring (if the roots are integers) or by completing the square. Factoring: x² - 5x + 6 = (x-2)(x-3) = 0, so x=2 or x=3. Completing the square converts the equation to vertex form and is how the quadratic formula itself was derived.

For ax² + bx + c = 0, the sum of roots = -b/a and the product of roots = c/a. These Vieta's formulas are useful for quickly checking your answers or forming a quadratic when you know its roots.

A linear equation (ax + b = 0) has exactly one solution and graphs as a straight line. A quadratic (ax² + bx + c = 0) can have 0, 1, or 2 real solutions and graphs as a parabola. Quadratics model accelerated or curved relationships, while linear models constant-rate relationships.