Robinson Tuition
Grade 6 · Higher · free GCSE maths worksheet

Quadratic formula

Solving ax^2 + bx + c = 0.

13 original questions, 45 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Solve x2 + 5x + 3 = 0

Give your solutions correct to 2 decimal places.

3 marks
Show answer
x = -0.70, x = -4.30
2

Here is the quadratic formula: x = −b ± √(b2 − 4ac)2a

Write down the values of a, b and c for 5x2 − 3x − 11 = 0, then substitute them into the formula. You do not need to simplify.

2 marks
Show answer
a = 5, b = -3, c = -11; x = (-(-3) ± √((-3)2 - 4 × 5 × (-11))) / (2 × 5)
3

Solve x2 - 4x - 9 = 0

Give your solutions correct to 2 decimal places.

3 marks
Show answer
x = 5.61, x = -1.61
4

Solve x2 − 6x − 5 = 0

Give your answers in the form p ± √q, where p and q are integers.

3 marks
Show answer
x = 3 + √14 or x = 3 - √14
5

Solve 2x2 + 7x - 3 = 0

Give your solutions correct to 2 decimal places.

3 marks
Show answer
x = 0.39, x = -3.89
6

Solve 3x2 − 7x − 4 = 0

Give your solutions correct to 2 decimal places.

3 marks
Show answer
x = 2.81, x = -0.47
7

Solve 3x2 - 5x - 7 = 0

Give your solutions correct to 2 decimal places.

3 marks
Show answer
x = 2.57, x = -0.91
8

Solve 3x(x − 2) = 5

Give your solutions correct to 3 significant figures.

4 marks
Show answer
x = 2.63, x = -0.633
9

A ball is thrown upwards from a platform. Its height, h metres, above the ground after t seconds is given by

h = 2 + 12t − 5t2

Work out how long it takes for the ball to hit the ground. Give your answer correct to 2 decimal places.

4 marks
Show answer
2.56 seconds
10

Solve x2 + 6x + 4 = 0

Give your solutions in the form a ± √b, where a and b are integers.

3 marks
Show answer
x = -3 + sqrt(5), x = -3 - sqrt(5)
11

Solve 2x2 - 8x + 5 = 0

Give your solutions in the form p ± √q2, where p and q are integers.

4 marks
Show answer
x = 4 + sqrt(6)2, x = 4 - sqrt(6)2
12
  1. Using the discriminant, explain why the equation x2 + 3x + 7 = 0 has no real solutions.
  2. Write down the number of real solutions of the equation x2 - 6x + 9 = 0. Give a reason for your answer.
4 marks
Show answer
a) b2 - 4ac = 32 - 4(1)(7) = -19, which is negative, so there are no real solutions b) One repeated real solution, because b2 - 4ac = 36 - 36 = 0
13

A rectangular lawn has length (x + 3) metres and width x metres.

The area of the lawn is 20 m2

  1. Show that x2 + 3x - 20 = 0
  2. Find the perimeter of the lawn. Give your answer correct to 3 significant figures.
6 marks
Show answer
a) Area = x(x + 3) = 20, so x2 + 3x = 20 and x2 + 3x - 20 = 0 b) 18.9 m

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

All Grade 6 topics