Algebra · sequences and functions

Explanation

• 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 $y=k$.

Explanation

• 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.

Explanation

• 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.

Explanation

• 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.

Explanation

• 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.

Explanation

• 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.

Explanation

• 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.

Explanation

• 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.