Equations & inequalities

Linear and simultaneous equations, the three ways to solve a quadratic, and solving linear inequalities.

9 min read

You must solve linear, simultaneous and quadratic equations confidently, and pick the quickest quadratic method for each question. Knowing the quadratic formula by heart is essential.

Solve linear equations by undoing operations in reverse order. For simultaneous equations, use elimination (match a coefficient, then add/subtract) or substitution.

Worked example

Solve $3 x + 2 y = 16$ and $5 x - 2 y = 8$ .

Add to eliminate $y$ : $8 x = 24 \Rightarrow x = 3$ . Then $9 + 2 y = 16 \Rightarrow y = 3.5.$

For a linear + quadratic pair, substitute the linear equation into the quadratic and solve the resulting quadratic.

Three methods, fastest first:

Factorising: best when it factorises neatly.
Completing the square: gives the turning point and exact roots.
Quadratic formula: always works: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ .

Worked example

Solve $2 x^{2} - 7 x + 3 = 0$ .

Factorising: $(2 x - 1) (x - 3) = 0 \Rightarrow x = \frac{1}{2} or x = 3.$

The discriminant $b^{2} - 4 a c$ tells you the number of real roots: positive ⇒ two, zero ⇒ one (repeated), negative ⇒ none.

Exam tip

Always glance for a factorisation first. On a non-calculator test, factorising a nice quadratic is far faster than grinding through the formula with surds.

Write $x^{2} + b x + c = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} + c$ . The bracket gives the turning point and lets you solve exactly.

Worked example

Solve $x^{2} + 6 x + 4 = 0$ by completing the square.

$(x + 3)^{2} - 9 + 4 = 0 \Rightarrow (x + 3)^{2} = 5 \Rightarrow x = - 3 \pm 5 .$

Solve linear inequalities like equations, with one rule: multiplying or dividing by a negative number reverses the inequality sign.

Worked example

Solve $5 - 2 x \leq 11$ .

$- 2 x \leq 6 \Rightarrow x \geq - 3$ (the sign flipped on dividing by $- 2$ ).

Forgetting to flip the inequality when dividing by a negative.
Losing the second root of a quadratic (every quadratic can have two solutions).
Sign errors in the formula (it is $- b$ on top, and $b^{2} - 4 a c$ under the root).

Equations & inequalities

Overview

Linear & simultaneous equations

Solving quadratics

Completing the square

Inequalities

Common mistakes