Algebra cheat sheet

Equivalent equations

Apply the same valid operation to both sides, while respecting restrictions such as division by zero.

3x + 4 = 19 → 3x = 15 → x = 5.

Distributive law

Multiplication distributes over addition. Factoring reverses this process by identifying common factors.

a(b + c) = ab + ac; 6x + 9 = 3(2x + 3).

Quadratics

A product is zero when at least one factor is zero. Factorisation can reveal roots of a quadratic equation.

x² − 5x + 6 = (x − 2)(x − 3), so the roots are 2 and 3.

Linear relationships

For y = mx + b, m is the change in y per unit change in x, and b is the y-intercept.

m = (y₂ − y₁)/(x₂ − x₁), when x₂ ≠ x₁.

Simultaneous equations

A solution of a system satisfies every equation. Substitution or elimination reduces the system while preserving the common solutions.

For 2x + y = 8 and x − y = 1, addition gives 3x = 9, so x = 3 and y = 2.

Inequalities

Adding the same quantity preserves an inequality. Multiplying or dividing by a negative quantity reverses its direction; dividing by an expression of unknown sign requires cases.

−2x < 6 gives x > −3.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac))/(2a). The discriminant determines the number of real roots.

A negative discriminant gives no real roots, though complex roots exist.

Index laws and restrictions

For suitable non-zero bases, aᵐaⁿ = aᵐ⁺ⁿ and a⁻ⁿ = 1/aⁿ. Real fractional powers require attention to domains and the chosen root.

2⁻³ = 1/8; a zero base cannot be used in 1/aⁿ.

Worked example: a rational equation

Solve x/(x − 1) = 2 with x ≠ 1. Multiplying by x − 1 gives x = 2x − 2, hence x = 2. Substitution gives 2/(2 − 1) = 2.

Write excluded values before multiplying by a denominator.

Common mistake: cancelling terms in a sum

Cancellation applies to common factors of a product, not arbitrary terms added in a numerator or denominator. Factor the whole expression before simplifying.

(x + 2)/x is not equal to 2; for x ≠ 0 it equals 1 + 2/x.

Check your understanding

Solve −3x ≥ 12.

x ≤ −4. Dividing by −3 reverses the inequality.

Curriculum connections

Sources