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Solve quadratic equations (ax² + bx + c = 0) using the quadratic formula. Shows discriminant, roots, and vertex.
Enter coefficients for: ax² + bx + c = 0
Quadratic Formula:
x = (-b ± sqrt(b² - 4ac)) / 2a
Discriminant D = b² - 4ac:
D > 0: two distinct real roots
D = 0: one repeated real root
D < 0: two complex conjugate roots
Vertex: (-b/2a, f(-b/2a))
A quadratic equation is a polynomial equation of degree 2, in the form ax² + bx + c = 0 where a ≠ 0. Its graph is a parabola.
The discriminant (b² - 4ac) determines the nature of roots. Positive: two real roots. Zero: one repeated root. Negative: two complex roots with no real solutions.
The vertex is the highest or lowest point of the parabola. Its x-coordinate is -b/(2a) and its y-coordinate is found by plugging that x back into the equation.
Only some quadratics factor neatly into integer factors. The quadratic formula works for all quadratic equations, including those with irrational or complex roots.
When the discriminant is negative, the square root produces an imaginary number. Complex roots come in conjugate pairs: a+bi and a-bi. They do not represent x-intercepts on a real number graph.