Factorise and solve quadratic equations by factorisation, the quadratic formula and completing the square, and plot and interpret quadratic graphs and their roots and turning points (Higher tier).

What this dot point is asking

Quadratics are the main Higher-tier algebra topic. WJEC asks you to factorise quadratic expressions, to solve quadratic equations by factorisation, by the quadratic formula and by completing the square, and to plot and interpret quadratic graphs, identifying their roots and turning points. A quadratic has the variable squared as its highest power, so it can have two solutions, and recognising which method fits a given equation is the key judgement. It appears on both components.

Factorising and solving

The fastest method, when it works, is to factorise into two brackets.

So $x^2 - 7x + 12 = 0$ factorises to $(x - 3)(x - 4) = 0$, giving $x = 3$ or $x = 4$. A special case is the difference of two squares, $x^2 - 9 = (x + 3)(x - 3)$.

The quadratic formula

When a quadratic does not factorise neatly, the formula always works.

The discriminant $b^2 - 4ac$ also tells you the number of roots: positive gives two, zero gives one (a repeated root), negative gives none.

Completing the square

Completing the square rewrites $x^2 + bx + c$ as $(x + \tfrac{b}{2})^2 + (c - \tfrac{b^2}{4})$. This form solves the equation by isolating the squared bracket and square rooting, and it directly gives the turning point of the graph: $y = (x + p)^2 + q$ has its minimum at $(-p, q)$. For $x^2 + 6x + 5$, completing the square gives $(x + 3)^2 - 4$, so the turning point is at $(-3, -4)$.

Quadratic graphs

The graph of a quadratic is a parabola, a symmetric U-shape (or an inverted U if the $x^2$ coefficient is negative).

Key features link to the algebra: the curve crosses the x-axis at the roots (the solutions of $ax^2 + bx + c = 0$), it crosses the y-axis at $c$, and its turning point lies on the line of symmetry, midway between the roots. Plotting a table of values and joining the points with a smooth curve produces the parabola, and reading where it crosses the x-axis gives approximate solutions.

A graph also lets you solve related equations by drawing a line. To solve $x^2 - 2x - 1 = 0$ from the graph of $y = x^2 - 2x - 3$, rearrange so the curve's equation appears: $x^2 - 2x - 3 = -2$, so draw the line $y = -2$ and read off where it meets the parabola. This "rearrange to match the drawn curve" technique is a common Higher question, turning a graph-reading task into an algebra one.

Choosing the right method

Picking the most efficient method saves time and reduces error. Try factorising first whenever the coefficients are small whole numbers, because it is fastest and exact. Reach for the quadratic formula when factorising fails or the question asks for decimal answers, since it always works. Use completing the square when you need the turning point, a minimum or maximum value, or an exact surd answer. Reading the command words ("give your answer to 2 decimal places" points to the formula; "find the minimum value" points to completing the square) tells you which tool the examiner expects.

Why this matters

Quadratics are the gateway from linear to non-linear algebra and appear across the Higher paper, in areas, in projectile-style problems and in the intersection of a line and a curve. The three solution methods cover every case, and recognising the most efficient one (factorise if it is clean, formula otherwise, complete the square for turning points) is a marked judgement. Surds reappear here whenever the discriminant is not a perfect square.