Quadratic Equations & the Discriminant

Three ways to solve ax² + bx + c = 0

Quadratics model projectile motion, area problems, and many optimisation setups before calculus. Master factoring, completing the square, and the quadratic formula.

Reading the discriminant

• Δ > 0: two distinct real roots.
• Δ = 0: one repeated real root (touches x-axis).
• Δ < 0: no real roots (complex conjugates).
• Sum of roots = −b/a; product = c/a (Vieta).

Worked example

Solve x² − 5x + 6 = 0. Factor: (x − 2)(x − 3) = 0 → x = 2 or 3. Δ = 25 − 24 = 1 > 0 confirms two roots.