quadratics

Quadratics are polynomials of degree two, typically written as f(x) = ax^2 + bx + c with a ≠ 0. The graph is a parabola opening upward when a > 0 and downward when a < 0. The axis of symmetry is x = −b/(2a) and the vertex is at (h, k) where h = −b/(2a) and k = f(h).

The roots of a quadratic are the values of x with f(x) = 0. The discriminant Δ = b^2 − 4ac determines the number and type of roots: Δ > 0 yields two distinct real roots; Δ = 0 yields a single real double root; Δ < 0 yields complex roots. Solutions can be found by factoring when possible, by completing the square, or by the quadratic formula x = (−b ± sqrt(Δ)) / (2a).

Factorization expresses f(x) as a(x − r1)(x − r2) when real roots exist, with r1 + r2 = −b/a and r1 r2 = c/a. The vertex form y = a(x − h)^2 + k, with h = −b/(2a) and k = f(h), emphasizes the location of the vertex. The parabola’s shape and key features link to the coefficients a, b, and c, and to the roots.

Quadratics appear across mathematics and its applications, including physics, engineering, economics, and modeling tasks such as projectile motion, optimization, and area problems.