Algebra worksheet — GCSE Higher

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

2.Solve 4x² − 9 = 0.

3.The point (5, −12) lies on the circle x² + y² = 169, which has centre (0, 0). Work out the equation of the tangent to the circle at (5, −12), giving your answer in the form y = mx + c.

4.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?

5.Two expressions are 4(x + 3) and 4x + 3. A student checks whether they are equivalent by substituting x = 2. Which statement correctly interprets the result?

6.The graph of y = f(x) has a minimum turning point at (3, 2). Write down the coordinates of the minimum turning point of the graph of y = f(x) + 5.

7.A rope of length L metres is cut into 6 equal pieces, and 4 metres is then removed from one piece. Write an expression, in metres, for the length of that piece after the cut.

8.A quadratic graph y = ax² + bx + c has its turning point on the y-axis. Which statement about its roots must be true?

9.f(x) = 3x − 2. Find f⁻¹(x).

10.Expand and simplify (x + 1)(x + 2)(x + 3).

11.Solve (2x + 1)/3 = 5

12.A candle is 30 cm tall and burns down at a steady rate of 1 cm per hour. Write down the function for the height of the candle y, in centimetres, after x hours.

13.Which expression means 'y squared, multiplied by 3'?

14.A line has equation y = −4x + 1. Work out the equation of the line perpendicular to it that passes through the point (0, 3).

15.By completing the square, show that the curve y = x² − 14x + 50 never crosses the x-axis. Which statement gives the correct reason?