Algebra worksheet — GCSE Higher

1.A sequence has the position-to-term rule n² − 3, where n is the position number. Work out the difference between the 6th term and the 5th term.

2.Write y = x² + 12x + 40 in the form (x + a)² + b, and hence write down the minimum value of y.

3.A geometric sequence has first term 3 and common ratio √3. Work out the position of the first term of the sequence that exceeds 30.

4.Work out the value of (x − 4)/2 + 3 when x = 10

5.Solve x² + 7x = 0.

6.Which expression is equivalent to 3(x + 4) − 2(x − 1)?

7.The equation x² + 2x − 5 = 0 can be solved using the iterative formula xₙ₊₁ = 5/(xₙ + 2). The starting value is x₀ = 1, so x₁ is the value after the formula has been used once. Work out x₂ correct to 2 decimal places.

8.A circle has equation x² + y² = 49. Work out the coordinates of the point(s) on the circle where the tangent is horizontal.

9.Solve the inequality 6.3x + 1.8 ≤ 40. Hence write down the greatest number with exactly one decimal place that satisfies the inequality.

10.To solve 6x − 4 = 2x + 20, Yusuf's first step is to subtract 2x from both sides. Work out what equation this gives.

11.Make x the subject of the formula y = 5x + 3.

12.A packet contains s sweets. The sweets are shared equally between f friends. Write an expression for the number of sweets each friend receives.

13.Work out the equation of the straight line through the points (−3, 4) and (1, −8).

14.The curve y = −x² + 6x − 5 has a maximum point. Use completing the square to find its coordinates.

15.A company's weekly profit, in £, is modelled by y = f(x), where x is the number of weeks since launch. The graph of y = f(x) has a maximum at (10, 45000). A rival company uses the same marketing strategy but starts trading 6 weeks later and has fixed costs £8000 higher every week, so its profit is modelled by y = f(x − 6) − 8000. In which week does the rival's maximum weekly profit occur, and what is it?