Algebra worksheet — GCSE Higher

1.Work out the value of the discriminant b² − 4ac for the equation 2x² + 3x − 2 = 0.

2.A theatre has rows of seats arranged so that row 1 has 12 seats, row 2 has 17 seats, row 3 has 22 seats and row 4 has 27 seats, with each row having 5 more seats than the row before. Work out an expression, in terms of n, for the number of seats in row n.

3.A curve has equation y = x³ − 4x. Work out the coordinates of the point where the curve crosses the y-axis.

4.An allotment is in the shape of a rectangle. Its length is 5 m more than its width, x metres, and its area is 20 m². This gives x² + 5x − 20 = 0, which can be solved using the iterative formula xₙ₊₁ = 20 ÷ (xₙ + 5). Taking x₀ = 2, so that x₁ is the value found after the formula has been used once, work out x₂ correct to 2 decimal places.

5.The numbers x and y satisfy x + y = 10 and xy = 21. Work out the pair of values.

6.Aisha saves money each week. In week 1 she has saved £15 in total, in week 2 she has saved £23 in total, in week 3 she has saved £31 in total and in week 4 she has saved £39 in total, with the total increasing by the same amount each week. Work out an expression, in terms of n, for the total amount she has saved, in pounds, after week n.

7.Solve the simultaneous equations 3x + y = 14 and x + y = 6. Work out the value of x.

8.The equation 3(2x − 1) = 4x + 9 is rearranged by expanding the brackets. Which of these is the correctly expanded equation?

9.A student solves the simultaneous equations 2x + y = 11 and x − y = 1 by elimination, adding the two equations together. Which of these is the correct result of that step?

10.A rectangle has area 24x + 18. Ffion factorises the area as 6(4x + 3). Work out the value of the area when x = 2, and decide whether 6 and (4x + 3) are correctly identified as factors of 24x + 18.

11.Solve the inequality 5x − 3 > 2x + 9.

12.A colony of bacteria doubles in number every hour. At 9am there are 5 bacteria in the colony. Work out how many bacteria there will be at 12 noon.

13.The equation x² − 6x + k = 0 has exactly one solution. Work out the value of k.

14.Work out the gradient of the straight line with equation 3y = 12 − 6x.

15.A runner's journey is shown on a distance-time graph: she runs 8 km in the first hour, rests for 30 minutes with no distance gained, then runs a further 4 km in the next 30 minutes. Work out her average speed for the whole journey, in kilometres per hour.