A Algebra · sequences and functions — notes on one page

A18 · Solve quadratic equations (including those requiring rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph

Definition. Specification A18: solve quadratic equations, including those that require rearrangement, algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph.

• Rearrange to $ax^2+bx+c=0$ first; expanding or clearing a fraction is often the first mark. Factorise when integer factors exist, then set each bracket to zero to get both roots.
• Higher tier: completing the square writes $x^2-8x+3$ as $(x-4)^2-13$, so $(x-4)^2=13$ and $x=4\pm\sqrt{13}$; with a coefficient of $x^2$ above $1$, take it out as a factor first.
• The quadratic formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ works for any quadratic; write the substitution line with the signs shown, because sign errors are what the reports blame.
• Give both solutions unless the context rules one out, in exact surd form when asked and otherwise rounded only at the end.
• From a graph, the solutions of $f(x)=0$ are the $x$-intercepts, and $f(x)=k$ is read where the curve meets the line $y=k$.

What the examiner wants. Signs under control. The June 2024 Paper 2H report says completing the square with a coefficient of $x^2$ greater than $1$ is always a struggle, with credit for taking out the factor of $2$ but very few completing it, and that where the quadratic formula was used it was invariably sign errors that led to incorrect solutions.

Exam tip. Write the substitution into the quadratic formula before evaluating; this is normally where the first method mark is earned.

A19 · Solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph

• A simultaneous solution is an ordered pair satisfying both equations; graphically it is an intersection point. For two linear equations, make one variable's coefficients equal or opposite, then add or subtract to eliminate it.
• Substitute the first value back to find the second.
• At Higher tier, when a line and a quadratic are solved together, substitute the linear expression into the quadratic, solve it, and find the matching second coordinate for every root.
• The examiner expects complete ordered pairs and a check in both original equations.
• Two $x$-values alone are incomplete because each belongs to a different point.

Exam tip. Write each solution as an ordered pair and substitute both coordinates into both equations when the tariff allows a checking mark.

A20 · Find approximate solutions to equations numerically using iteration [Higher only]

• Higher tier only. Iteration rewrites an equation as $x=g(x)$ and repeatedly applies $x_{n+1}=g(x_n)$ from the stated starting value.
• Each new output becomes the next input. Keep the calculator's full stored value and round only values you are asked to record, because premature rounding can change later iterates.
• Convergence is indicated when successive values agree to the required accuracy, although not every rearrangement converges.
• The examiner expects the requested named iterate or a justified approximate root, with enough intermediate values to show the process.
• Substitute the final approximation into the original equation as a reasonableness check.

Exam tip. Use the calculator answer key to carry the full previous iterate into the next substitution.

A21 · Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution

Definition. Specification A21: translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution.

• Start by defining the unknown in words, with units: let $x$ cm be the width. Write every other quantity as an expression in $x$ from the information given, such as $2x$, $2x-6$ or $x+3$.
• The equation comes from the one fact that links them:
• a perimeter, an angle sum of $90^\circ$, $180^\circ$ or $360^\circ$, a total number of counters, or two expressions for equal sides. Solve it by the usual balancing steps, then answer the question that was asked, which is rarely the value of $x$ itself: substitute back to find the angle, the side length, the area or the ratio.
• Schemes give a process mark for the expressions, one for the equation, one for solving and one for using the solution, so each stage must be visible.
• A root that gives a negative length or a non-integer count is rejected with a reason. Label the final answer with what it represents.

What the examiner wants. An equation before anything else. The November 2024 Paper 2H report says the first mark was for forming an appropriate equation from the diagram, either summing all the sides to $72$ or equating the two equal sides, the second for the process to isolate the terms in $x$, and that most who reached that point solved correctly.

Exam tip. Define the variable before forming the equation and finish with a sentence interpreting every required quantity.

A22 · Solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph

• Solve a linear inequality like an equation, reversing the inequality sign only when multiplying or dividing by a negative number.
• On a number line, use a filled endpoint when equality is included and an open endpoint for a strict inequality.
• At Higher tier, represent two-variable inequalities by drawing each boundary and testing a point to choose the region; strict boundaries are dashed.
• For a quadratic inequality, find the roots and test the intervals they define, because the required values may lie inside or outside the roots.
• The examiner expects the correct endpoint style, shading and notation as well as the algebraic boundary values.

Exam tip. For a graphical answer, the boundary style and the direction of shading are separate marking points.

A23 · Generate terms of a sequence from either a term-to-term or a position-to-term rule

• A term-to-term rule produces each new term from the preceding term or terms, so begin with every stated starting value and apply the operations in the given order.
• A position-to-term rule gives a term directly from its position $n$; for the first terms substitute $n=1,2,3,\ldots$ unless the question explicitly defines another starting index.
• Multi-step, alternating and Fibonacci-type rules require intermediate terms to be shown because a later term may depend on more than one earlier value.
• The examiner expects the terms in order and usually gives method credit for correct substitutions or repeated operations even if a later arithmetic error occurs.

Exam tip. Write each intermediate term because a correct recurrence method can still earn marks after one arithmetic slip.

A24 · Recognise and use triangular, square and cube numbers, arithmetic progressions, Fibonacci type sequences, quadratic sequences, simple geometric progressions (r^n, r rational > 0 or a surd) and others

• Square and cube numbers have forms $n^2$ and $n^3$, while triangular numbers have form $\dfrac{n(n+1)}{2}$. An arithmetic progression has a constant first difference and a quadratic sequence has constant second differences.
• Fibonacci-type sequences form later terms from preceding terms.
• At Higher tier, a geometric progression has a constant positive rational or surd multiplier and can be written using powers such as $r^n$.
• Check several consecutive steps before identifying a sequence, then use the defining rule consistently.
• The examiner expects the named structure to be supported by differences, ratios or a valid term relationship rather than by visual resemblance alone.

Exam tip. Show a short difference table or two equal consecutive ratios to justify the sequence type.

A25 · Deduce expressions to calculate the nth term of linear and quadratic sequences

• For a linear sequence with common difference $d$, start with $dn$ and adjust the constant so the expression gives the first term when $n=1$.
• At Higher tier, a quadratic sequence $an^2+bn+c$ has constant second difference $2a$.
• Find $a$, subtract the values of $an^2$ from the original terms, and identify the linear rule left behind to obtain $b$ and $c$.
• The examiner expects an expression in $n$, not merely the next term.
• Verify the rule against at least two supplied terms; this detects a shifted index or an incorrect constant before the final answer.

Exam tip. For a quadratic sequence, display the first and second differences because they secure the method for finding the $n^2$ coefficient.