AQA A-Level Mathematics: Quadratics, Simultaneous Equations and Inequalities (7357)

This guide covers B3 (Quadratic Functions and Their Graphs), B4 (Simultaneous Equations) and B5 (Linear and Quadratic Inequalities), from Section B: Algebra and Functions in AQA A-level Mathematics (7357), first teaching September 2017. This content is prescribed by the Department for Education and shared across every exam board offering A-level Mathematics, not set independently by AQA.

Scope of this guide

At A-level, Section B runs from B1 (laws of indices) through B11. B1-B9 end with transformations of graphs; the two further statements are B10 (decomposing rational functions into partial fractions) and B11 (use of functions in modelling, including consideration of limitations and refinements of the models). (B1-B9 alone is the AS range, not the A-level one.) This resource covers B3, B4 and B5, a natural cluster centred on quadratics: working with the quadratic function itself, solving it simultaneously against a linear equation, and handling inequalities involving it. B1-B2 (indices and surds), B6-B9 (polynomial manipulation, graphs of functions, composite/inverse functions, transformations), B10 (partial fractions) and B11 (functions in modelling) are left for separate resources.

Syllabus coverage

AQA A-LEVEL MATHEMATICS (7357) — SECTION B: ALGEBRA AND FUNCTIONS

• B3 — work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations, including solving quadratic equations in a function of the unknown
• B4 — solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation
• B5 — solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions; express solutions through correct use of “and” and “or”, or through set notation; represent linear and quadratic inequalities graphically

How to approach it

Treat the discriminant, b² − 4ac, as the single tool that connects all three of these sub-topics. In B3, it tells you directly how many real roots a quadratic has: two distinct real roots if b² − 4ac > 0, one repeated root if b² − 4ac = 0, and no real roots if b² − 4ac < 0. In B4, when you substitute a linear equation into a quadratic to solve simultaneously, the resulting quadratic’s discriminant tells you whether the line intersects the curve twice, touches it once (tangent), or misses it entirely. In B5, sketching the graph of a quadratic inequality relies on knowing where (and whether) the curve crosses the x-axis — which is exactly what the discriminant and the roots determine. Revising these three sub-topics through that single connecting idea, rather than as separate procedures, is significantly more efficient than learning each in isolation.

B4 names both methods – solving “by elimination and by substitution, including one linear and one quadratic equation” – so you need to be able to use either. For the linear-quadratic case, substitution is often the most convenient route in practice: rearrange the linear equation to isolate one variable, substitute it into the quadratic, and solve the resulting single-variable quadratic. Equating two expressions for y (which eliminates y) is equally valid and gives the same quadratic; the choice is a matter of convenience, not correctness. For B5, be precise with “and” versus “or”: a solution like −2 < x < 3 (an “and” condition, x greater than −2 and less than 3) describes a bounded region between two roots, typically for a quadratic inequality opening upwards that is less than zero; a solution like x < −2 or x > 3 describes the unbounded regions outside two roots, typically for the same quadratic set greater than zero.

Worked example: solving a quadratic inequality graphically

Solve x² − x − 6 > 0.

Step 1: find the roots (where the expression equals zero)
        x^2 - x - 6 = 0
        (x - 3)(x + 2) = 0
        x = 3 or x = -2

Step 2: sketch the graph
        upward-opening parabola crossing the x-axis at x = -2 and x = 3

Step 3: identify where the graph is above the x-axis (> 0)
        this occurs outside the two roots

Step 4: write the solution using "or"
        x < -2 or x > 3

Sketching the graph as an explicit step, even briefly, is what prevents the single most common error in this sub-topic: writing the solution the wrong way round (as a bounded “and” region instead of the correct unbounded “or” region, or vice versa).

Key terms to define precisely

Discriminant — the expression b² − 4ac from the quadratic formula, whose sign determines the number and type of real roots a quadratic equation has. Repeated root — a single value of x where the quadratic graph touches the x-axis tangentially (discriminant equals zero) rather than crossing it. Completing the square — rewriting a quadratic in the form a(x + p)² + q, useful for finding a quadratic’s turning point and for solving equations without using the quadratic formula directly. Set notation — a formal way of expressing a solution set, such as {x : −2 < x < 3}, as an alternative to writing the inequality in words with “and”/“or.” Precision about which of these terms a question is asking for — a numeric answer, a graph sketch, or a set-notation expression — avoids losing marks for providing the right mathematical content in the wrong required format.

Common mistakes

Solving a quadratic inequality algebraically without sketching the graph first, leading to the solution being written the wrong way round (“and” instead of “or”, or vice versa). Believing only one method is acceptable for a linear-quadratic simultaneous pair: B4 requires both elimination and substitution, and what loses marks is an algebra slip (for example, not substituting back to find the matching y-values), not the choice of method. Forgetting to check the discriminant before assuming a quadratic has real roots at all. Dropping a solution when solving an inequality involving a fraction or bracket, by treating it exactly like a linear inequality without accounting for how the quadratic terms change the shape of the solution set.

Quick revision checklist

• Learn what each sign of the discriminant (positive, zero, negative) tells you about a quadratic’s roots.
• Practise solving simultaneous linear-quadratic pairs by both substitution and elimination, as B4 requires.
• Always sketch the graph before writing the final solution to a quadratic inequality.
• Be precise about “and” (bounded, between two roots) versus “or” (unbounded, outside two roots) in inequality solutions.

Official syllabus

AQA A-level Mathematics (7357) specification, first teaching 2017 — aqa.org.uk/7357.