Leaving Cert Algebra (Higher Level): Equations, Factorising & Inequalities

Q: Why is algebra so important for Leaving Cert Maths?

Algebra is the language of the entire course. Calculus, complex numbers, functions and coordinate geometry all rely on confident algebraic manipulation, so improving algebra lifts your whole grade.

Q: How do I solve simultaneous equations in the Leaving Cert?

Use elimination (add or subtract the equations to cancel a variable) or substitution (rearrange one equation and substitute into the other). Both are accepted methods.

Q: What is the quadratic formula?

The quadratic formula gives the solutions of ax² + bx + c = 0 and is provided on the exam formula and tables booklet.

Algebra is the foundation of Leaving Cert Higher Maths, covering solving equations, factorising, simultaneous equations, quadratics and inequalities. It underpins almost every other topic on Paper 1.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

Algebra appears on Paper 1 and is used inside nearly every other topic on both papers.
The quadratic formula solves any quadratic equation when factorising is not possible.
Simultaneous equations can be solved by substitution or elimination.
Strong algebra skills directly improve your performance in calculus, complex numbers and functions.

Algebra explained

What is a variable?

A variable is a symbol (like $x$ , $y$ , or $a$ ) that represents a number. $3x$ means “3 multiplied by $x$ ” (like 3 apples). Algebra is about writing and solving problems using these symbols.

Examples: “A number plus 7” → $x + 7$ “Double a number” → $2x$

Like and Unlike terms

Like terms have the same variable and the same power. Example: $2x$ and $5x$ are like terms. Unlike terms cannot be combined. Example: $3x$ and $2y$ are different. $3x$ and $3x^2$ are different

Simplifying:

$2x + 3x = 5x$

$7y - 4y + 4x = 3y +4x$

$4x + 2y + 5x - y = 9x + y$

Multiplying and Expanding

When multiplying variables:

$x * x = x^2$

$2x * 3x = 6x^2$

$5x * 3y = 15xy$

Expanding brackets means multiplying everything inside:

$2(x + 3) = 2x + 6$

$(x + 2)(x + 5) = x^2 + 7x + 10$

This is an introduction. The full Algebra lessons, worked solutions and practice questions are available to members. Start your free trial.

Key formulas

Name	Formula	Description
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	Finds roots of quadratic equation ax² + bx + c = 0
Discriminant	$\Delta = b^2 - 4ac$	Determines nature of roots: Δ>0 two real roots, Δ=0 one root, Δ<0 no real roots
Sum of Roots	$\alpha + \beta = -\frac{b}{a}$	Sum of roots of quadratic equation ax² + bx + c = 0
Product of Roots	$\alpha \beta = \frac{c}{a}$	Product of roots of quadratic equation ax² + bx + c = 0
Difference of Squares	$a^2 - b^2 = (a+b)(a-b)$	Factorising difference of two squares
Sum of Cubes	$a^3 + b^3 = (a+b)(a^2-ab+b^2)$	Factorising sum of two cubes

Worked examples

Worked example 1

What are the roots of the function $x^2 + 2x - 3$ if $x + 3$ is a factor.

Factor Theorem

Answer:

$x=-3$ & $x=1$

Worked example 2

Solve for $x$ : $2x + 5 = 11$

Subtract 5 from both sides: $2x = 6$ . Then divide by 2: $x = 3$ .

Answer:

Where students lose marks

Sign errors when expanding brackets or moving terms across the equals sign.
Forgetting the ± when taking a square root.
Not checking both solutions of a quadratic in the original context.

Frequently asked questions

Why is algebra so important for Leaving Cert Maths?