GCSE higher: quadratic formula, iteration and inequalities

Quadratic formula, iteration, inequalities, explained

Many quadratic equations will not factorise neatly, so two other methods are taught. Completing the square rewrites x² + 6x − 4 = 0 as (x + 3)² − 13 = 0, so x + 3 = ±√13 and x = −3 ± √13. The quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a, solves any equation written as ax² + bx + c = 0. Some equations need rearranging first, such as x² + 3x = 4 − x, which must become x² + 4x − 4 = 0 before a, b and c can be read off. Answers are given as exact surds or rounded as the question asks.

In simultaneous equations where one equation is linear and the other quadratic, the usual approach is substitution. Take y = x + 1 and x² + y² = 25. Replace y in the second equation to get x² + (x + 1)² = 25, which simplifies to x² + x − 12 = 0, so (x + 4)(x − 3) = 0. That gives x = −4 or x = 3, and the linear equation then gives y = −3 or y = 4. The solutions must be written as pairs, here (−4, −3) and (3, 4). Graphically, these are the two points where a straight line crosses a circle.

Iteration finds approximate solutions by repeating a calculation. Rearrange x³ − 3x − 5 = 0 into x = ∛(3x + 5), start with x₀ = 2, and feed each answer back in. To two decimal places the values run 2.22, 2.27, 2.28, 2.28 and settle near 2.28. Pupils use the calculator’s answer key to repeat the step quickly. To show that a root is 2.28 to two decimal places, they can also show a change of sign: the expression is negative at 2.275 and positive at 2.285, so a root lies between.

A linear inequality in two variables, such as y > 2x − 1, is shown by drawing the boundary line and shading a region. The line is dashed when the inequality is strict (< or >) and solid when it includes equality (≤ or ≥). Testing a point such as (0, 0) shows which side is correct. Several inequalities together mark out a region, and schools differ on whether to shade the wanted region or the unwanted one, so follow the instruction on the question.

A quadratic inequality in one variable is solved by finding where the quadratic equals zero and then sketching the curve. To solve x² − x − 6 < 0, factorise to (x − 3)(x + 2), so the curve crosses the axis at −2 and 3. It is below the axis between them, so −2 < x < 3. For x² > 9 the answer has two parts, x < −3 or x > 3. Set notation writes the solution as a set, for example {x : −2 < x < 3}, meaning “the set of all x such that x is between −2 and 3”.

Words your child will hear

Completing the square

Rewriting x² + bx + c as (x + b/2)² plus or minus a number, to solve the equation or find the turning point.

Quadratic formula

A formula that gives the solutions of ax² + bx + c = 0 from the numbers a, b and c.

Surd

A root that cannot be written exactly as a whole number or fraction, such as √13. Leaving it in surd form keeps the answer exact.

Iteration

Repeating a rule, using each answer as the next input, to home in on a solution.

Strict inequality

One that uses < or > and does not include the boundary value.

Boundary line

The line, such as y = 2x − 1, that separates the region that fits an inequality from the rest of the graph.

Set notation

A way of writing a solution as a set, such as {x : x < −3}, read as “the set of x such that x is less than −3”.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

• Solve quadratic equations, including those that require rearrangement, algebraically by completing the square and by using the quadratic formula
• Solve 2 simultaneous equations in 2 variables (linear/quadratic) algebraically
• Find approximate solutions to equations numerically using iteration
• Solve linear inequalities in 2 variables and represent the solution set on a graph
• Solve quadratic inequalities in 1 variable
• Represent the solution set of an inequality using set notation

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Common mistakes and how to help

How to help at home

• Put the answers back in. After solving any equation, substitute each solution into the original. For the worked example, 2 × 0.85² + 3 × 0.85 − 4 comes out very close to 0. This habit catches sign slips.
• Try a calculator iteration. Type any positive number, then repeat “add 2, then square root” using the answer key. The results settle at 2. Ask why: 2 is the number where √(2 + 2) = 2 again.
• Turn a budget into an inequality. Pears cost 60p and apples 40p, and you can spend at most £3. With x pears and y apples, working in pence, that is 60x + 40y ≤ 300. List some combinations that fit. On a graph, the whole-number points inside the region are all the possible shopping baskets.
• Test points on a number line. For x² − x − 6 < 0, try x = −3, 0 and 4 in the expression and see which ones work. It shows why the answer is a single range, and the habit helps on every inequality.
• Explain which method and why. Give your teenager three quadratics and ask which method they would choose for each, and why. Spotting the quickest method is a real skill.

Related topics

Everything in Key Stage 4 maths