Worked examples

Solve $x^2 + 6x + 2 = 0$ by completing the square. Give your answers in exact (surd) form.

Solve a quadratic exactly by completing the square — the method examiners want when 'give exact answers' appears.

Solve the simultaneous equations $y = x + 3$ and $x^2 + y^2 = 29$.

One linear, one quadratic — substitute, solve, and don't forget to find BOTH coordinates.

Simplify fully $\dfrac{x^2 - 9}{2x^2 + 5x - 3}$.

Factorise top and bottom, then cancel — the standard route through an algebraic fraction.

Rationalise the denominator and simplify $\dfrac{5 + \sqrt{3}}{2 - \sqrt{3}}$.

Multiply by the conjugate to clear a surd from the bottom of a fraction.

Solve $3x^2 - 5x - 1 = 0$, giving your answers to 2 decimal places.

When it won't factorise, use the formula — carefully with the signs and the rounding.

Solve $x^2 - x - 6 > 0$.

Solve the equation, sketch the parabola, then read the region — the sketch is what stops sign errors.

Prove algebraically that the difference between the squares of any two consecutive odd integers is divisible by 8.

Turn consecutive odd numbers into algebra, then expose the factor that proves divisibility.

In triangle $OAB$, $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. Point $P$ lies on $OA$ with $OP:PA=2:1$, and point $Q$ lies on $OB$ with $OQ:QB=2:1$. Prove that $PQ$ is parallel to $AB$, and state the ratio $PQ:AB$.

Express two routes with vectors, then prove parallel lines by finding a positive scalar multiple.

The equation $x^3+x-5=0$ is rearranged as $x_{n+1}=\sqrt[3]{5-x_n}$. Starting with $x_0=1.5$, calculate $x_1$ to $x_5$ and hence estimate the root to 3 decimal places.

Apply a recurrence accurately and use stabilising decimal places to estimate a cubic root.

Points $A$, $B$, $C$ and $D$ lie on a circle. $AB$ is a diameter, and $A$ and $D$ lie in the same segment cut off by chord $BC$. Angle $BAC=(3x+4)^\circ$ and angle $BDC=(5x-10)^\circ$. Find $x$ and angle $ABC$.

Link same-segment angles, a diameter, and a triangle angle sum in one algebraic chain.

$f(z) = z^3 - 5z^2 + 16z - 30$. Given that $1 + 3i$ is a root of $f(z) = 0$, find the other two roots.

Use the conjugate pair to build a real quadratic factor, then compare coefficients to find the last root.

$\mathbf{M} = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 3 \\ 1 & 0 & 2 \end{pmatrix}$. Find $\det \mathbf{M}$ and hence find $\mathbf{M}^{-1}$.

Determinant, minors, cofactors, transpose, divide — the full 3×3 inverse with a check at the end.

Prove by induction that, for all $n \in \mathbb{Z}^+$, $f(n) = 5^n + 8n + 3$ is divisible by 4.

Basis, assumption, inductive step via f(k+1) − f(k), and the full conclusion the mark scheme insists on.

Use the method of differences to show that $\displaystyle\sum_{r=1}^{n} \dfrac{2}{(r+2)(r+4)} = \dfrac{n(7n+25)}{12(n+3)(n+4)}$.

Partial fractions first, then write out enough terms to see exactly which ones survive the cancellation.

The curve $C$ has polar equation $r = 4 + 3\cos\theta$, $0 \le \theta < 2\pi$. Find the exact area enclosed by $C$.

Half the integral of r squared — expand, use the double-angle identity, and integrate over one full turn.

Find the particular solution of $\dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} + 2y = 4x$ for which $y = 0$ and $\dfrac{dy}{dx} = 1$ when $x = 0$.

Auxiliary equation, complementary function, particular integral, then the boundary conditions — in that order.

Use algebra to find the set of values of $x$ for which $\dfrac{6}{x+2} \le x - 3$.

Never multiply through by an expression of unknown sign — collect to one fraction, find critical values, test the intervals.

Use the substitution $t = \tan\dfrac{\theta}{2}$ to solve $3\sin\theta - 4\cos\theta = 4$ for $0 \le \theta < 2\pi$. Give any non-exact answers to 3 significant figures.

Substitute t = tan(θ/2), solve in t — then check θ = π, the root the substitution can never find.

The parabola $C$ has equation $y^2 = 12x$. The point $P(12, 12)$ lies on $C$. The normal to $C$ at $P$ meets $C$ again at the point $Q$. Find the coordinates of $Q$.

Gradient of the curve, perpendicular gradient, then solve with the parabola — one root is the point you started from.

A network has vertices $A, B, C, D, E, F$ and weighted edges $AB = 4$, $AC = 2$, $BC = 1$, $BD = 5$, $CD = 8$, $CE = 10$, $DE = 2$, $DF = 6$, $EF = 3$. Use Dijkstra's algorithm to find the shortest route from $A$ to $F$ and state its length.

Order of labelling, final values, working values — then trace the route back from the destination.

A carpenter makes chairs and tables. Each chair needs 2 hours of cutting and 2 hours of finishing; each table needs 3 hours of cutting and 1 hour of finishing. Each week at most 24 hours of cutting and 16 hours of finishing are available. The profit is £30 per chair and £40 per table. Formulate this as a linear programming problem and use a graphical method to find how many chairs and tables should be made each week to maximise the profit.

Define the variables in words, write the objective and constraints, then test the vertices of the feasible region.

A project consists of the activities below, with durations in days. $A$ (3) and $B$ (5) have no predecessors. $C$ (4) and $D$ (5) depend on $A$. $E$ (2) depends on $B$ and $C$. $F$ (4) depends on $D$ and $E$. $G$ (3) depends on $E$. Find the minimum completion time, the critical path and the total float of each non-critical activity.

Forward pass takes the largest, backward pass takes the smallest — then float picks out the critical activities.