Quadratic Equations — Objectives | Mathematics Department

Solving Quadratic Equations

1. Solve a quadratic equation by factorising and applying the null factor law.
2. Solve a quadratic equation of the form $x^2 = k$ or $(x - a)^2 = k$.
3. Solve rational equations that reduce to a quadratic after clearing fractions.
4. Complete the square to solve a quadratic equation.
5. Solve a quadratic equation using the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
6. Give exact (surd) solutions to quadratic equations where appropriate.

7. Identify the $y$-intercept of a quadratic function as $f(0)$.
8. Find the $x$-intercepts (roots) of a quadratic by setting $f(x) = 0$.
9. Determine the axis of symmetry and the vertex (maximum or minimum) of a quadratic.
10. Sketch a quadratic from its key features (intercepts, vertex, direction of opening).
11. Convert between expanded, factorised and vertex forms of a quadratic.

12. Solve optimisation problems modelled by a quadratic (e.g. maximum area, maximum profit, projectile motion).
13. Solve a system consisting of a linear and a quadratic equation algebraically.
14. Use the discriminant $b^2 - 4ac$ to determine the nature of the roots (two distinct, one repeated, none).
15. Solve quadratic inequalities using a sketch or sign-table.