Quadratic functions and equations in O-Level Additional Mathematics (4049): completing the square, the discriminant and nature of roots, quadratic inequalities, and equations reducible to quadratic form

The most reused algebra in the course

Quadratics sit near the centre of the Algebra strand of Additional Mathematics (SEAB 4049), and the techniques in this module are reused constantly: completing the square to find a turning point, the discriminant to count roots, sign reasoning for inequalities, and substitution to reduce harder equations to quadratics. The discriminant in particular reappears as the tangency test in coordinate geometry, so this topic pays off well beyond its own questions.

This module covers four outcomes: completing the square, the discriminant and nature of roots, quadratic inequalities, and equations reducible to quadratic form. This overview ties the four dot points together; each has its own worked answers and practice.

Completing the square

The completing the square outcome rewrites $ax^2 + bx + c$ as $a(x - h)^2 + k$. This form gives the vertex $(h, k)$, the line of symmetry $x = h$, and the maximum or minimum value $k$ directly. It is the reliable route to a parabola's turning point and to the range of the function.

The discriminant and nature of roots

The discriminant and nature of roots outcome uses $b^2 - 4ac$ to classify the roots without solving: positive gives two distinct real roots, zero gives one repeated root, and negative gives no real roots. Questions often impose a root condition to find an unknown constant.

Quadratic inequalities

The quadratic inequalities outcome solves inequalities such as $x^2 - x - 6 > 0$. You rearrange to zero, factorise to find the roots, then use a sketch or sign reasoning: for an upward parabola the expression is negative between the roots and positive outside, so the inequality direction picks the interval.

Equations reducible to quadratic form

The equations reducible to quadratic form outcome handles equations that become quadratic after a substitution. Letting $u$ stand for a repeated expression (for example $u = x^2$ in $x^4 - 5x^2 + 4 = 0$, or $u = \sqrt{x}$ in a surd equation) turns the equation into one you can factorise, after which you revert to the original variable and keep only the valid solutions.

How this module is examined

• Both papers, all questions. Paper 1 and Paper 2 (each 2 hours 15 minutes, 90 marks, 50 percent) cover the full syllabus, and you answer every question.
• State conditions precisely. For root questions, write the discriminant condition explicitly before solving, and remember equal roots use strict equality.
• Revert and check after substitution. After solving the quadratic in $u$, substitute back to find $x$ and discard any values that are invalid (for example a negative value when $u = \sqrt{x}$).

Check your knowledge

Short questions across the four outcomes. Work them with full method, then check the solutions.

1. Express $x^2 - 6x + 11$ in completed-square form and state its minimum value. (3 marks)
2. State the nature of the roots of $2x^2 + 3x + 5 = 0$ using the discriminant. (2 marks)
3. Solve the inequality $x^2 - x - 6 \le 0$. (3 marks)
4. Solve $x^4 - 5x^2 + 4 = 0$. (3 marks)