Solve ax² + bx + c = 0 using the quadratic formula. Find real and complex roots, discriminant, vertex, and graph properties.
ax² + bx + c = 0
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Solve any quadratic equation instantly using the quadratic formula. Enter coefficients a, b, and c to find roots (real or complex), the discriminant, vertex coordinates, and axis of symmetry. Perfect for algebra, physics, and engineering.
The quadratic formula x = (-b ± √(b²-4ac)) / 2a solves equations of the form ax² + bx + c = 0. The discriminant (b²-4ac) determines the nature of roots: positive gives two real roots, zero gives one repeated root, negative gives two complex conjugate roots.
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Step-by-step solution breakdown
The discriminant (b²-4ac) reveals the nature of roots: when positive, there are two distinct real roots; when zero, there's one repeated real root; when negative, there are two complex conjugate roots. The larger the discriminant, the further apart the roots are.
Complex roots occur when the discriminant is negative, meaning there's no real number solution. These roots are expressed as a + bi and a - bi, where i is the imaginary unit (√-1). Complex roots always come in conjugate pairs.
The vertex is the highest or lowest point on the parabola, located at x = -b/(2a). If a > 0, the parabola opens upward and the vertex is the minimum. If a < 0, it opens downward and the vertex is the maximum.
If a = 0, the equation becomes linear (bx + c = 0), not quadratic. A quadratic equation by definition must have an x² term with a non-zero coefficient.