Solve quadratic equations (ax² + bx + c = 0) with step-by-step solutions, discriminant analysis, and graph visualization.
Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
Discriminant: Δ = b² - 4ac
Vertex: h = -b/(2a), k = f(h)
Axis of Symmetry: x = -b/(2a)
Nature of Roots based on Discriminant:
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It always has two solutions (roots), which may be real or complex numbers.
The discriminant (Δ = b² - 4ac) determines the nature of the roots. If positive, there are two real roots. If zero, there's one repeated real root. If negative, the roots are complex numbers with imaginary parts.
The vertex is the point where the parabola changes direction. For a quadratic in standard form, the vertex coordinates are (h, k) where h = -b/(2a) and k is the value of the function at h. It represents the maximum or minimum point.
To graph: find the vertex, plot the y-intercept (where x=0, y=c), find the x-intercepts (roots), plot a few additional points, and draw a smooth curve through them. The parabola opens upward if a>0 and downward if a<0.
Complex roots occur when the discriminant is negative. They are expressed in the form a ± bi, where i is the imaginary unit (√-1). Complex roots always come in conjugate pairs and represent points where the parabola doesn't intersect the x-axis.
