Solves quadratic equations ax² + bx + c = 0 using the quadratic formula and visualizes with graphs. Useful for math learning, homework, and exam preparation.
A quadratic equation is an equation with the highest degree of 2, in the form ax² + bx + c = 0, where a, b, c are constants and a ≠ 0. Examples include x² - 5x + 6 = 0, 2x² + 3x - 2 = 0. Quadratic equations graph as parabolas, and where the graph intersects the x-axis are the solutions (roots) of the equation.
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. This formula can solve any quadratic equation. For example, in x² - 5x + 6 = 0 with a=1, b=-5, c=6, substituting gives x = (5 ± √(25-24)) / 2 = (5 ± 1) / 2, so x = 3 or x = 2. This formula was perfected by 16th century mathematicians.
The discriminant Δ = b² - 4ac determines the nature of the roots. If Δ > 0, two distinct real roots; if Δ = 0, repeated root (two equal real roots); if Δ < 0, two complex roots. For example, x² - 4x + 4 = 0 has Δ = 16 - 16 = 0, so x = 2 (repeated root). x² + x + 1 = 0 has Δ = 1 - 4 = -3 < 0, so it has complex roots.
The graph of y = ax² + bx + c is a parabola. If a > 0, it opens downward (concave up); if a < 0, it opens upward (concave down). The vertex x-coordinate is -b/2a, and the axis of symmetry is x = -b/2a. Where the graph intersects the x-axis are the roots, and the number of roots can be determined by the discriminant.
When the two roots of ax² + bx + c = 0 are α and β, the equation factors as a(x - α)(x - β) = 0. For example, since x² - 5x + 6 = 0 has roots 2 and 3, it factors as (x - 2)(x - 3) = 0. From the relationship between roots and coefficients, α + β = -b/a and αβ = c/a.
Quadratic equations are used in physics for projectile motion (ball trajectories), architecture for arch design, economics for profit maximization, and engineering for optimization problems. For example, for a ball thrown from height h with y = -5t² + 20t + h, the time when the ball hits the ground (y=0) can be found by solving a quadratic equation.