National 5 Maths: Quadratic Formula, Completing the Square and Sketching

Quadratic functions run through both National 5 Mathematics question papers. On the non-calculator paper you are expected to complete the square, read off a turning point and sketch a parabola; on the calculator paper the quadratic formula is a regular question, often inside a problem where you first have to form the equation yourself. This material trains the quadratic skills of the National 5 course beyond simple factorising.

The quiz has thirteen questions. Two ask you to solve an equation with the quadratic formula and round the roots to one decimal place, one of them after rearranging the equation so that one side is zero. Two ask you to complete the square with a unitary $x^{2}$ coefficient. Others ask you to state the coordinates and nature of the turning point of $y = k (x + p)^{2} + q$ , give the equation of the axis of symmetry, find the roots and the $y$ -intercept you need for a sketch, work out $k$ in $y = k x^{2}$ from a point on the graph, and find $k$ , $p$ and $q$ from a turning point and one other point. The discriminant appears in an equal-roots problem, and a rectangle question asks you to solve with the formula and explain why one root is rejected. The wrong options are built from the slips that cost marks most often: sign errors in the turning point, dividing by $a$ instead of $2a$, forgetting to rearrange, and rounding too early.

The flashcards summarise the method for completing the square, the quadratic formula, the language used to describe the roots (two real and distinct roots, one repeated real root or two equal real roots, no real roots), the turning point and axis of symmetry of $y = k (x + p)^{2} + q$ , and how to decide between a minimum and a maximum turning point.

The printable written work has eight longer problems to answer by hand: solving with the formula, completing the square and explaining why a graph never meets the $x$ -axis, preparing two sketches with every key point, finding equations from the features of a graph, using the discriminant, and a context problem about the height of a ball where both roots have a meaning. Each asks for full working, as the exam does.

In the oral practice exam an examiner asks one question at a time, asks you to explain your method, and ends with short feedback on what to practise next.

The material suits S4 pupils preparing for National 5 and anyone revising before starting Higher, where the same skills are extended.

Solve a quadratic equation with the quadratic formula and round the roots correctly
Complete the square for a quadratic with a unitary x² coefficient
State the turning point, its nature and the axis of symmetry of y = k(x + p)² + q
Find the roots and y-intercept needed to sketch a quadratic graph
Determine the equation of a parabola of the form y = kx² or y = k(x + p)² + q from its features
Use the discriminant to describe the roots and to find an unknown coefficient

Practice material written by Zestly, based on the National 5 Mathematics course specification (Qualifications Scotland, formerly SQA; version 3.0, May 2023), algebraic skills: completing the square, quadratic functions, the quadratic formula and the discriminant.

Sample question

Solve the equation $2x^2 - 3x - 4 = 0$ using the quadratic formula. Provide the roots rounded to one decimal place.

$x = -0.8, 2.3$
$x = -0.9, 2.4$
$x = -1.1, 2.6$
$x = -0.7, 2.2$

See the answer

$x = -0.9, 2.4$

Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=2, b=-3, c=-4$, we get $x = \frac{3 \pm \sqrt{9 - 4(2)(-4)}}{4} = \frac{3 \pm \sqrt{41}}{4}$. This yields $x \approx 2.35$ and $x \approx -0.85$, which round to $2.4$ and $-0.9$.

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