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.

Differentiate $y = (3x^2 + 1)^5$ with respect to $x$.

Differentiate a 'function inside a function' — name the inside, then multiply the derivatives.

Differentiate $y = x^2 e^x$ with respect to $x$.

Differentiate a product of two functions — then factorise, because the mark scheme wants it tidy.

Find $\displaystyle\int x e^x \, dx$.

Choose u to be the part that simplifies when you differentiate it — then it's just the formula.

Find the first three terms, in ascending powers of $x$, of the binomial expansion of $(1 + 3x)^{-2}$.

The general binomial series — mind the signs and remember to raise the whole x-term to each power.

Express $\dfrac{3x + 1}{(x - 1)(x + 2)}$ in partial fractions.

One fraction per linear factor, then substitute the clever values of x to pick off each constant.

Solve $2\sin^2 x + \sin x - 1 = 0$ for $0 \le x < 360^\circ$.

Spot the hidden quadratic in sin x, solve it, then find every angle in the range.

The curve $x^2+xy+y^2=7$ passes through $(1,2)$. Find the equation of the tangent to the curve at this point.

Differentiate every term in an implicit curve, collect the derivative, then build the tangent.

A curve has parametric equations $x=t+\dfrac1t$ and $y=t-\dfrac1t$, where $t>0$. Find a Cartesian equation of the curve and the equation of the tangent when $t=2$.

Eliminate the parameter, then use parametric differentiation to find a tangent on the same curve.

Use the trapezium rule with four equal strips to estimate $\displaystyle\int_0^2 \dfrac{1}{1+x}\,dx$. Give your estimate to 3 decimal places and state, with a reason, whether it is an overestimate or an underestimate.

Build a trapezium estimate from exact ordinates, then use curvature to decide whether it is high or low.

Use the Newton–Raphson method with $x_0=2$ to solve $x^3-2x-5=0$. Calculate $x_1$, $x_2$ and $x_3$, and give the root to 3 decimal places.

Form the Newton–Raphson recurrence from a function and derivative, then iterate without premature rounding.