Quadratic Equations and Quadratic Functions
Quadratic equations and quadratic functions are examined together because the algebra of roots and the geometry of the parabola are two views of one object. SPM questions ask for the nature of the roots, equations formed from given roots, completed-square form, graphs and quadratic inequalities.
Roots and the discriminant
For ax² + bx + c = 0, the discriminant D = b² − 4ac decides the roots: D > 0 gives two distinct real roots, D = 0 gives two equal real roots, D < 0 gives no real roots. If the roots are α and β, then α + β = −b/a and αβ = c/a, so the equation is x² − (α+β)x + αβ = 0.
Completing the square
Write f(x) = a(x−h)² + k. The vertex is (h, k), the axis of symmetry is x = h, and k is the minimum value when a > 0 or the maximum when a < 0.
Worked example
f(x) = 2x² − 8x + 5 = 2(x² − 4x) + 5 = 2[(x−2)² − 4] + 5 = 2(x−2)² − 3. The vertex is (2, −3), the minimum value is −3, and the axis is x = 2. The discriminant is 64 − 40 = 24 > 0, so the graph cuts the x-axis twice, consistent with a minimum below the axis.
Traps
Forgetting to factor out a before completing the square; dropping the minus sign in α + β = −b/a; writing D ≥ 0 when the question specifies equal roots; reversing an inequality when a < 0.