Algebra - Solving equations and inequalities: solve quadratic equations algebraically by factorising; find approximate solutions using a graph (Foundation tier)

GCSE Pearson Edexcel Mathematics Foundation: A18 Quadratic Equations

Overview

Quadratic equations contain an x^2 term and can model a curved relationship rather than a constant rate of change. For Pearson Edexcel GCSE Mathematics qualification 1MA1, A18 at Foundation asks you to solve quadratic equations algebraically by factorising and to find approximate solutions from a graph. A11, A12, A17 and A21 connect this work to quadratic graphs, function language, linear solving and forming equations from situations. The secure Foundation target is the monic form x^2+bx+c=0: find two numbers that add to b and multiply to c, then use the zero-product rule. 1 2

You will also need to recognise a root on a graph, distinguish an exact algebraic answer from a graph estimate, and decide whether an answer makes sense in context. Rearrangement examples show why the right-hand side must be zero before using the zero-product rule. Non-monic factorisation, completing the square and the quadratic formula sit beyond this Foundation A18 guide. Apart from the explicitly cited November 2024 question, examples and allocations are original illustrations, not official mark schemes. Work neatly, substitute to check, and give every valid solution unless a context rejects one.

Key Concepts

Recognise a quadratic

A quadratic equation is commonly written in the general form ax^2+bx+c=0, where $a
e0$. Must memorise / not supplied on checked 2026 Foundation sheet: this is the general form; at Foundation A18, use the monic case a=1, x^2+bx+c=0. Pearson's 2026 Foundation aid provides geometry, trigonometry, compound-interest and probability formulae, but no quadratic factorising identities or solving formula. 3

Find the factor pair

For a monic quadratic, expand two brackets:

Must memorise / not supplied on checked 2026 Foundation sheet:
(x+p)(x+q)=x^2+(p+q)x+pq.

Must memorise / not supplied on checked 2026 Foundation sheet: to factorise x^2+bx+c, choose p and q so that p+q=b and pq=c. The product tells you which signs are alike or different; the sum checks the middle term. For x^2-2x-15, -5\times3=-15 and -5+3=-2, so
(x-5)(x+3)=0.

Use the zero-product rule

Must memorise / not supplied on checked 2026 Foundation sheet: if AB=0, then A=0 or B=0. This zero-product rule works only because the product is zero. If AB=12, neither factor need be zero, so you cannot set A=0 or B=0.

Solve each bracket as a linear equation rather than relying only on a “flip the sign” mnemonic. From x-5=0, add 5, so x=5. From x+3=0, subtract 3, so x=-3. These are the solutions. A command word matters: factorise asks for (x-5)(x+3), while solve asks for x=5 and x=-3.

This exact example appeared as Question 26 on Pearson's November 2024 Foundation Paper 1F. The official scheme, available in the supplied hosted copy, awards 3 marks: M1 for a valid factorisation attempt meeting a coefficient condition, M1 for (x+3)(x-5), then A1 for 5,-3. 4 That is evidence for this question, not a universal “one mark per algebra line” rule.

Mini worked example, ordinary factors. Solve x^2+x-6=0. The pair 3 and -2 has product -6 and sum 1.
(x+3)(x-2)=0.
Set each factor to zero: x+3=0\Rightarrow x=-3, or x-2=0\Rightarrow x=2.

Mini worked example, rearrange first. Solve x^2+x=6. Subtract 6 from both sides to make zero on one side: x^2+x-6=0. The factorising above gives x=-3 or x=2. The formal Foundation guidance limits A18 factorising to x^2+bx+c=0, so rearrangement here is useful preparation and linked algebra, not a reason to assume every rearranged quadratic is a Foundation factorising question. 2

Keep zero roots and recognise special factors

Must memorise / not supplied on checked 2026 Foundation sheet: a common factor can make a quadratic product. For x(x-4)=0, do not divide by x, because that could lose the solution x=0. Use the zero-product rule: x=0 or x-4=0, so x=4.

Must memorise / not supplied on checked 2026 Foundation sheet: the difference of two squares is x^2-d^2=(x-d)(x+d). For x^2-25=0, write (x-5)(x+5)=0; hence x=5 or x=-5.

Must memorise / not supplied on checked 2026 Foundation sheet: a repeated factor (x-r)^2=0 gives one repeated root, x=r. For x^2-6x+9=0, (x-3)^2=0, so the root is x=3 twice. State it as one solution, or as a repeated root if asked.

Mathematical Relationships

A factorised equation and its graph tell the same story. Must memorise / not supplied on checked 2026 Foundation sheet: for y=f(x), a root is an x-value for which f(x)=0, so it is an x-axis intercept. Must memorise / not supplied on checked 2026 Foundation sheet: solving f(x)=k means finding where the curve y=f(x) meets the horizontal line y=k.

There may be two, one repeated, or no real roots. An upward-opening curve that crosses the x-axis twice has two real roots. A curve that touches the x-axis at its turning point has a repeated root, and that touching point is a root. A curve entirely above the axis has no real roots. Algebra explains x^2+1=0: for real x, x^2\geq0, therefore x^2+1\geq1, so it can never equal zero.

Failure to find an integer factor pair does not prove that there are no real roots. For example, x^2-2x-1=0 has two non-integer real roots visible on its graph. A non-negativity argument or a suitable graph can justify a no-real-roots claim at this level. Ask yourself why a curve above the x-axis could still intersect a higher horizontal line.

The practical time guide is sensible but not a board rule. Each 1MA1 paper is 90 minutes and 80 marks, so 90\div80=1.125 minutes per mark. That is roughly one minute per mark with time kept for checking, not a speed requirement. 1 Questions can be linked multi-part problems worth 4 to 6 marks. They reward connecting known algebra, graphs and geometry, rather than a mysterious extra method. If told use the graph, read the given graph; if told show your working, show the method, not just the final roots.

Foundation boundary

Non-monic quadratics that need additional factorisation, such as 2x^2+7x+3=0, are Higher-only for this guide. Completing the square as a method for solving and use of the quadratic formula are also Higher-only. Do not try to memorise a formula as a required Foundation method. Pearson's specification and teaching guidance distinguish the Foundation factorising target from these Higher methods. 1 2

Practical Applications

Area is a common route from a situation to a quadratic. Must memorise / not supplied on checked 2026 Foundation sheet: for a rectangle, A=lw. Here A is area, l is length and w is width. If adjacent lengths are x and x+2, its area is x(x+2).

Mini worked example, rectangle. A rectangle has area 48\text{ cm}^2, length x+2 cm and width x cm. Form the equation
x(x+2)=48.
Rearrange: x^2+2x-48=0. Since 8\times(-6)=-48 and 8+(-6)=2,
(x+8)(x-6)=0.
Thus x=-8 or x=6. A length cannot be negative, so reject -8. The rectangle is 6 cm by 8 cm. Check: 6\times8=48.

When units change, square the conversion factor for area. Must memorise / not supplied on checked 2026 Foundation sheet: 1\text{ m}=100\text{ cm}. Must memorise / not supplied on checked 2026 Foundation sheet: 1\text{ m}^2=10000\text{ cm}^2, because 100\text{ cm}\times100\text{ cm}=10000\text{ cm}^2. Units help decide whether a numerical solution is physically possible.

There is no required practical experiment for this mathematics topic. A graph exercise is mathematical practice, not a science-style required practical. Use pencil, graph paper and a ruler for axes and reading guides, and draw the quadratic itself as a smooth curve.

Graph Skills

To plot y=x^2-2x-1, calculate reliable points before drawing. At x=-1,0,1,2,3, the corresponding y values are 2,-1,-2,-1,2. Plot (-1,2), (0,-1), (1,-2), (2,-1) and (3,2). Use an even scale, label both axes and read the numbered grid carefully. A scale might make each small square represent 0.2, not 1.

Draw one smooth U-shaped curve through the points. Do not join the points with ruler segments: a quadratic is curved between plotted values. Where this curve crosses y=0, read roots of approximately -0.4 and 2.4 to 1 decimal place. These graph answers are approximations. Algebra can sometimes give exact integer roots; a graph gives the precision supported by its scale.

For the same function, solve f(x)=2 by drawing the horizontal line y=2. It meets the curve at x=-1 and x=3, so the solutions are -1 and 3. A quick algebra check is x^2-2x-1=2, hence x^2-2x-3=0, which factorises to (x-3)(x+1)=0. On a graph, always read the intersection x-coordinates, not their y-coordinates, unless the question asks for coordinates.

Retrieval Checkpoints

Try these without notes. They are original practice prompts, not Pearson questions or allocations.

1. Factorise and solve x^2-7x+12=0.
2. Explain why the zero-product rule cannot be used directly on (x-1)(x+4)=8.
3. Solve x(x+5)=0 without dividing by x.
4. State the number and type of real roots of x^2-6x+9=0.
5. For y=f(x), what does an x-axis intercept mean?
6. A positive rectangle side is x and the other is x+3. If its area is 28\text{ cm}^2, what are the side lengths?

Answers. For 1, (x-3)(x-4)=0, so x=3,4. For 2, the product is 8 rather than zero, so neither factor is forced to be zero. For 3, x=0 or x+5=0, giving 0,-5. For 4, there is one repeated real root, x=3. For 5, it is a root: an input where f(x)=0. For 6, x^2+3x-28=0=(x+7)(x-4), so x=4 after rejecting -7; the sides are 4 cm and 7 cm.

Ten-minute Podcast

Listen to a warm British female tutor and answer before she does. 00:00–01:00, introduction: what Foundation A18 covers and how to listen actively. 01:00–06:00, core concepts: rearrange, factorise, solve, check, then distinguish root types and graph intersections. 06:00–08:00, exam tips: authentic three-mark scheme, command words, timing and common errors. 08:00–09:00, recall quiz: factor-pair conditions, bracket signs, zero roots, repeated roots and the correct horizontal line. 09:00–10:00, summary: retrieve the complete method and plan a later revision attempt. The full transcript is included in the podcast_script field.

References

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