24 questions working through all three methods of solving a quadratic, plus roots and turning points from the graph.

📈 Quadratics: factorise, complete the square, formula

Maths · GCSE Higher · 24 questions · ~45 min

MathsUKwww.geekhero.co.uk

Almost every Higher paper asks you to solve a quadratic, and the mark is often lost not on the algebra but on choosing the wrong method for the quadratic in front of you. This sheet takes the three methods in turn and then mixes them. It starts with factorising, including ax² + bx + c and the difference of two squares; moves to completing the square, and to reading the turning point straight out of the completed form; then to the quadratic formula, where the marks go missing on −b when b is already negative and on dividing the whole numerator by 2a. The last section gives you quadratics with no instruction, so you have to decide for yourself. Roots, intercepts and turning points from the graph are included too, because the graphical and algebraic halves of this topic are examined together.

1.Factorise fully 5x + 5y − 5
(a)5(xy − 1)
(b)5(x + y + 1)
(c)5(x + y − 1)
(d)5(x + y − 5)
2.A quadratic graph has roots at x = −3 and x = 5, and it crosses the y-axis at (0, −15). Work out the equation of the curve in the form y = (x − a)(x − b).
(a)y = (x − 3)(x − 5)
(b)y = (x − 3)(x + 5)
(c)y = (x + 3)(x + 5)
(d)y = (x + 3)(x − 5)
3.Expand and simplify (x + 1)(x + 2)(x + 3).
(a)x³ + 6x² + 11x + 5
(b)x³ + 6x² + 6x + 6
(c)x³ + 5x² + 11x + 6
(d)x³ + 6x² + 11x + 6
4.Solve 2x² = 18.
(a)x = 9 or x = −9
(b)x = 4 or x = −4
(c)x = 6 or x = −6
(d)x = 3 or x = −3
5.Work out the value of the discriminant b² − 4ac for the equation 2x² + 3x − 2 = 0.
(a)13
(b)−7
(c)25
(d)22
6.A rectangular photo has length (x + 3) cm and width x cm. Its area is 40 cm². Work out the value of x.
(a)x = −8
(b)x = 5
(c)x = 5 or x = −8
(d)x = 4
7.The equation x² − 6x + k = 0 has exactly one solution. Work out the value of k.
(a)k = −9
(b)k = 9
(c)k = 3
(d)k = 36
8.Factorise x² − 64.
(a)(x − 4)(x + 16)
(b)(x − 8)(x − 8)
(c)(x + 8)(x + 8)
(d)(x − 8)(x + 8)
9.By completing the square, show that the curve y = x² − 14x + 50 never crosses the x-axis. Which statement gives the correct reason?
y = x² − 14x + 50
(a)The turning point is above the x-axis at (7, −1)
(b)50 is positive, so y is always positive
(c)The discriminant is positive, so y is never zero
(d)(x − 7)² + 1 is at least 1 for every x, so y is never 0
10.The graph of y = x² − 5x + 6 crosses the x-axis at two points. By factorising, work out the x-coordinates of these two points.
y = x² − 5x + 6
(a)x = −1 and x = −6
(b)x = 2 and x = −3
(c)x = 2 and x = 3
(d)x = −2 and x = −3
11.Write y = x² + 12x + 40 in the form (x + a)² + b, and hence write down the minimum value of y.
y = x² + 12x + 40
(a)−6
(b)36
(c)4
(d)40
12.Solve x² − 100 = 0.
(a)x = 10 or x = −10
(b)x = 10
(c)x = 100 or x = −100
(d)x = 50
13.Which of these values of x is a solution of x² + 2x − 15 = 0?
(a)x = 3
(b)x = 5
(c)x = 15
(d)x = −3
14.A table shows y = x² − 6x + 5 at these points (x, y): (0, 5), (1, 0), (2, −3), (3, −4), (4, −3), (5, 0), (6, 5). Using the symmetry shown, write down the x-coordinate of the turning point.
y = x² − 6x + 5
(a)x = 1
(b)x = 5
(c)x = 6
(d)x = 3
15.By completing the square, find the turning point of the curve y = 3x² + 12x + 7.
y = 3x² + 12x + 7
(a)x = −2, y = 5
(b)x = −4, y = −41
(c)x = 2, y = −5
(d)x = −2, y = −5
16.Simplify (5x² + 3x − 2) − (2x² − x + 5)
(a)3x² + 4x + 3
(b)3x² + 2x + 3
(c)7x² + 2x + 3
(d)3x² + 4x − 7
17.Subtract, giving your answer as a single fraction in its simplest form: 4/(x − 1) − 2/(x + 3)
(a)(2x + 11)/((x − 1)(x + 3))
(b)(2x + 14)/((x − 1)(x + 3))
(c)2/((x − 1)(x + 3))
(d)(2x − 10)/((x − 1)(x + 3))
18.Expand and simplify (x − 3)²
(a)x² + 6x + 9
(b)x² − 9
(c)x² − 3x + 9
(d)x² − 6x + 9
19.Which of these quadratic graphs does NOT cross the x-axis at all?
(a)y = (x − 2)² + 3
(b)y = (x + 4)(x − 1)
(c)y = (x − 2)(x + 3)
(d)y = x² − 9
20.Simplify 5a/6 − a/3
(a)7a/6
(b)2a/3
(c)4a/3
(d)a/2
21.A ball's height, h metres, t seconds after being thrown follows h = (t − 1)(9 − t). Given that the ball is at ground level at t = 1 and t = 9, work out at what time t the ball reaches its maximum height, using symmetry.
(a)t = 4
(b)t = 8
(c)t = 10
(d)t = 5
22.Simplify 2/(x + 1) + 3/(x − 2), giving your answer as a single fraction.
(a)5/(2x − 1)
(b)(5x + 1)/(x² − x − 2)
(c)(5x − 1)/(x² + x − 2)
(d)(5x − 1)/(x² − x − 2)
23.Expand and simplify 4(2x − 1) − 3(x + 2) − 5x
(a)−10
(b)5x − 10
(c)2
(d)−7
24.A rectangular garden has length (x + 7) m and width (x − 7) m. Work out an expression for the area of the garden, giving your answer in its simplest form.
(a)2x
(b)x² + 49
(c)x² − 14x − 49
(d)x² − 49

What is on this worksheet?

The sheet holds 24 questions drawn from the MathsUK bank — the content area covered: Algebra (statements A18, A11, A4). It is pitched at GCSE Higher and takes about 45 minutes to work through in full. It is built for independent practice, with full answers at the end for self-marking.

How to use the sheet well

Print it or open it on screen — both work. Printing is A4; the screen view fits phones and tablets.
Do all 24 questions before checking — about 45 minutes is the guide, but there is no time pressure.
Check the answers — press “Show answers” or print the answer page separately.
Redo the questions you got wrong — twice as effective as doing 24 fresh ones.
“New questions” — builds a fresh sheet on the same statements, so you can practise again without repeats.

Why this sheet helps

MathsUK worksheets use questions graded by difficulty and a fair spread of correct-answer positions (the answer is not always (a)) — so the student really has to think about each question rather than guess a pattern. Every question is tagged to a DfE content statement and checked before it enters the bank. The answers come with a step-by-step explanation, not just a value — so a wrong answer becomes a lesson.

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