Quadratic Equations Worksheet — GCSE — Intermediate
Quadratic Equations worksheet for Key Stage 4 / GCSE (Ages 14-16). Intermediate level maths practice, aligned to the UK National Curriculum. Print-ready with answer key included.
Mastering Quadratic Equations
Key Stage 4 / GCSE (Ages 14-16) - Intermediate Level
Instructions: This worksheet will help you develop your skills in solving quadratic equations. You will encounter a variety of problems requiring different techniques such as factorisation, completing the square, and using the quadratic formula. Ensure to show all your workings clearly.
Solving Quadratics by Factorisation
Factorise each quadratic equation and solve for x. Show all steps clearly.
1. x² - 5x + 6 = 0
2. x² + 3x - 10 = 0
3. x² - 7x + 12 = 0
4. x² + 4x - 21 = 0
Completing the Square
Rewrite each quadratic equation by completing the square and solve for x. Show all steps.
1. x² + 6x + 5 = 0
2. x² - 4x - 5 = 0
3. x² + 8x + 10 = 0
4. x² - 2x - 15 = 0
Using the Quadratic Formula
Solve the following quadratic equations using the quadratic formula. Show all workings.
1. 2x² - 4x - 6 = 0
2. x² + 5x + 6 = 0
3. 3x² - 2x - 1 = 0
4. x² - 3x - 10 = 0
Word Problems Involving Quadratics
Read each problem carefully, form a quadratic equation, and solve it. Provide all necessary steps.
1. A rectangular garden has an area of 48 square metres. If the length is 2 metres more than the width, find the dimensions of the garden.
2. The product of two consecutive integers is 56. Find the integers.
Graphing Quadratic Functions
For each quadratic function, determine the vertex and the axis of symmetry, then sketch the graph.
1. y = x² - 4x + 3
2. y = -x² + 2x + 1
Identifying the Nature of Roots
Determine the nature of the roots for each quadratic equation using the discriminant. State whether the roots are real and distinct, real and equal, or complex.
1. x² - 3x + 2 = 0
2. x² + 2x + 5 = 0
3. x² - 4x + 4 = 0
Quadratic Inequalities
Solve the following quadratic inequalities and represent the solution on a number line.
1. x² - 5x + 6 < 0
2. x² + 2x - 8 > 0
Real-life Application of Quadratics
Discuss how quadratic equations are used in real life. Provide examples and explain their significance.
Write an essay discussing the applications of quadratic equations in areas such as physics, engineering, and finance. [10 marks]
Activity 1: x = 2, 3 ; x = -5, 2 ; x = 3, 4 ; x = -7, 3
Activity 2: x = -1, -5 ; x = 5, -1 ; x = -5, -2 ; x = 5, -3
Activity 3: x = 3, -1 ; x = -2, -3 ; x = 1, -1/3 ; x = 5, -2
Activity 4: 6m x 8m ; 7 and 8
Activity 5: Vertex: (2, -1), Axis: x = 2 ; Vertex: (1, 2), Axis: x = 1
Activity 6: Real and distinct ; Complex ; Real and equal
Activity 7: (2, 3) ; x < -4 or x > 2
Activity 8: Essays will vary
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