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.

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.