The quadratic function for Edexcel IGCSE Further Pure Mathematics: Methods and Worked Examples

The quadratic function sits at the heart of algebra, and in the edexcel igcse further pure mathematics specification it goes well beyond what you covered at standard IGCSE level

You already know how to solve quadratic equations. What this course adds is a deeper toolkit: completing the square for any quadratic, using the discriminant to classify roots before you find them, and working with the relationships between roots and coefficients to build new equations from old ones. These skills turn up everywhere in the exam, from pure algebra questions to calculus and coordinate geometry problems where a quadratic appears as an intermediate step.

This set of edexcel igcse further pure mathematics the quadratic function revision notes covers the full section. If you can handle everything on this page confidently, you have one of the most versatile tools in the course locked down.

Manipulation of quadratic expressions

Factorising

Factorising a quadratic expression means writing it as a product of two linear factors. For ax² + bx + c where a = 1, you look for two numbers that multiply to c and add to b. When a is not equal to 1, you can use grouping or inspection.

Worked example: Factorise 6x² + 7x - 3.

We need two numbers that multiply to 6 x (-3) = -18 and add to 7. Those numbers are 9 and -2.

Rewrite: 6x² + 9x - 2x - 3 = 3x(2x + 3) - 1(2x + 3) = (3x - 1)(2x + 3).

Completing the square

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This form immediately tells you the vertex of the parabola (at x = -p, y = q) and whether the parabola opens upward (a > 0) or downward (a < 0).

Worked example: Write 2x² - 12x + 5 in completed square form.

Factor out the coefficient of x² from the first two terms: 2(x² - 6x) + 5.

Complete the square inside the bracket: x² - 6x = (x - 3)² - 9.

Substitute back: 2[(x - 3)² - 9] + 5 = 2(x - 3)² - 18 + 5 = 2(x - 3)² - 13.

The minimum value of y is -13, occurring when x = 3.

Why completing the square matters beyond the quadratic itself. In the edexcel igcse further pure mathematics exam, completing the square is not just a standalone skill. It turns up when you need to find the vertex of a parabola for a graph-sketching question, when you need to write a denominator in a form that reveals an asymptote, and when you need to determine the range of a quadratic function. Getting fast at this technique pays dividends across the entire specification.

Roots of a quadratic equation

The quadratic formula

For ax² + bx + c = 0, the roots are given by x = (-b +/- \u221a(b² - 4ac)) / (2a). This formula is on the formulae sheet, so you do not need to memorise it, but you do need to apply it accurately under exam pressure.

Worked example: Solve 3x² - 5x + 1 = 0, giving your answers to 3 significant figures.

Here a = 3, b = -5, c = 1.

Discriminant: b² - 4ac = 25 - 12 = 13.

x = (5 +/- \u221a13) / 6.

x = (5 + 3.6055...) / 6 = 1.43 (3 s.f.) or x = (5 - 3.6055...) / 6 = 0.232 (3 s.f.).

The discriminant

The discriminant is the expression b² - 4ac. It tells you about the nature of the roots without solving the equation:

Value of b² - 4ac	Nature of roots
Positive (b² - 4ac > 0)	Two distinct real roots
Zero (b² - 4ac = 0)	Two equal real roots (one repeated root)
Negative (b² - 4ac < 0)	No real roots

Worked example: Find the values of k for which the equation 2x² + kx + 8 = 0 has equal roots.

For equal roots, b² - 4ac = 0.

k² - 4(2)(8) = 0, so k² = 64, giving k = 8 or k = -8.

Discriminant questions often appear as "show that" problems. The exam might give you a line and a curve and ask you to show they do not intersect. You substitute the line equation into the curve equation to get a quadratic, then show the discriminant is negative. Keep your algebra tidy: a sign error in the discriminant calculation will undermine the entire argument.

Functions of the roots of a quadratic equation

This is the part of the quadratic function edexcel igcse further pure mathematics topic that goes furthest beyond standard IGCSE. If the roots of ax² + bx + c = 0 are alpha and beta, then:

alpha + beta = -b/a (sum of roots)
alpha x beta = c/a (product of roots)

These relationships let you work with the roots without actually finding them, which is powerful when the roots are irrational or complex.

Worked example: finding symmetric functions of roots

The roots of 2x² - 7x + 4 = 0 are alpha and beta. Find the value of alpha² + beta².

Sum of roots: alpha + beta = 7/2.

Product of roots: alpha x beta = 4/2 = 2.

Using the identity alpha² + beta² = (alpha + beta)² - 2(alpha x beta):

alpha² + beta² = (7/2)² - 2(2) = 49/4 - 4 = 49/4 - 16/4 = 33/4.

Worked example: forming a new equation

The roots of x² - 5x + 3 = 0 are alpha and beta. Find the equation whose roots are alpha + 1 and beta + 1.

From the original equation: alpha + beta = 5 and alpha x beta = 3.

New sum of roots: (alpha + 1) + (beta + 1) = alpha + beta + 2 = 5 + 2 = 7.

New product of roots: (alpha + 1)(beta + 1) = alpha x beta + alpha + beta + 1 = 3 + 5 + 1 = 9.

The new equation is x² - 7x + 9 = 0.

Worked example: roots involving reciprocals

The roots of 3x² + 2x - 5 = 0 are alpha and beta. Find the equation whose roots are 1/alpha and 1/beta.

Sum: alpha + beta = -2/3. Product: alpha x beta = -5/3.

New sum: 1/alpha + 1/beta = (alpha + beta) / (alpha x beta) = (-2/3) / (-5/3) = 2/5.

New product: 1/(alpha x beta) = 1/(-5/3) = -3/5.

The new equation is x² - (2/5)x + (-3/5) = 0. Multiplying through by 5: 5x² - 2x - 3 = 0.

Common symmetric function	Expression using sum and product
alpha² + beta²	(alpha + beta)² - 2(alpha x beta)
alpha³ + beta³	(alpha + beta)³ - 3(alpha x beta)(alpha + beta)
1/alpha + 1/beta	(alpha + beta) / (alpha x beta)
(alpha - beta)²	(alpha + beta)² - 4(alpha x beta)

Edexcel IGCSE Further Pure Mathematics practice questions

Test your understanding of the quadratic function with these igcse 4PM1 the quadratic function problems.

Write 3x² + 18x + 20 in the form a(x + p)² + q and state the minimum value of the expression.
The equation kx² + (2k + 1)x + (k - 1) = 0 has two distinct real roots. Find the range of values of k.
The roots of 2x² - 3x - 1 = 0 are alpha and beta. Find the value of alpha³ + beta³.
The roots of x² + 4x + 1 = 0 are alpha and beta. Find the equation whose roots are alpha² and beta².

Solutions

Question 1: 3x² + 18x + 20 = 3(x² + 6x) + 20 = 3[(x + 3)² - 9] + 20 = 3(x + 3)² - 27 + 20 = 3(x + 3)² - 7. The minimum value is -7.

Question 2: For two distinct real roots, b² - 4ac > 0. (2k + 1)² - 4(k)(k - 1) > 0. 4k² + 4k + 1 - 4k² + 4k > 0. 8k + 1 > 0. k > -1/8. Also k cannot be 0 (otherwise the equation is linear), so k > -1/8 and k is not equal to 0.

Question 3: Sum: alpha + beta = 3/2. Product: alpha x beta = -1/2. alpha³ + beta³ = (alpha + beta)³ - 3(alpha x beta)(alpha + beta) = (3/2)³ - 3(-1/2)(3/2) = 27/8 + 9/4 = 27/8 + 18/8 = 45/8.

Question 4: alpha + beta = -4, alpha x beta = 1. New sum: alpha² + beta² = (alpha + beta)² - 2(alpha x beta) = 16 - 2 = 14. New product: (alpha x beta)² = 1. The equation is x² - 14x + 1 = 0.

The quadratic function is one of those areas where the edexcel igcse further pure mathematics notes you compile should include a table of standard identities for symmetric functions of roots. Having those identities at your fingertips eliminates the thinking time that costs marks under exam conditions. The edexcel igcse further pure mathematics explained approach here is to build fluency with these relationships so that forming new equations from given roots becomes mechanical rather than creative.

For more edexcel igcse further pure mathematics practice questions on quadratics and every other section of the specification, use edexcel igcse further pure mathematics revision notes on the Green Bridge CBT platform to work through exam-style problems with instant feedback.