Quadratic equations and functions

Builds on

Required prerequisites, all the way down: 21

You will learn to

• Solve quadratic equations by factorising, completing the square and the quadratic formula, and derive the formula.
• Use the discriminant to determine the number of real roots.
• Sketch y = ax² + bx + c with its vertex, axis of symmetry and intercepts, and solve optimisation problems.

Definition

• A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0.
• The discriminant is Δ = b² − 4ac.
• The vertex of a parabola is its highest or lowest point.

Theorem

• Quadratic formula: x = (−b ± √Δ)/(2a) when Δ ≥ 0.
• There are two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.
• Vieta: the roots r and s satisfy r + s = −b/a and rs = c/a.
• Completing the square: ax² + bx + c = a(x + b/(2a))² − Δ/(4a).

Common misconceptions

• Dividing both sides by x and losing the root x = 0.
• Misapplying the zero-product property, e.g. concluding from (x − 1)(x − 2) = 6 that x − 1 = 6 or x − 2 = 6.
• Thinking Δ < 0 means there are no solutions in any number system; there are two complex solutions.

Leads to

Optional discoveries

• Introductory calculus The vertex of a parabola is where its slope is zero, an idea that calculus extends to all differentiable functions.

Learning pathways

• Conventional K–12 progression
• Accelerated secondary progression

Further reading

• I. M. Gelfand and A. Shen, Algebra (1993)