Quadratic Equations Worksheet — GCSE — Advanced
Quadratic Equations worksheet for Key Stage 4 / GCSE (Ages 14-16). Advanced level maths practice, aligned to the UK National Curriculum. Print-ready with answer key included.
Advanced Quadratic Equations Mastery
Key Stage 4 / GCSE (Ages 14-16) - Advanced Level
Instructions: This worksheet is designed to challenge your understanding of quadratic equations. You will solve complex problems, analyse graphs, and apply your knowledge to real-world scenarios. Ensure you show all your workings clearly.
Solving Quadratic Equations
Solve each quadratic equation using the quadratic formula. Show all your workings clearly and simplify your answers where possible.
1. x² - 4x - 5 = 0
2. 2x² + 3x - 2 = 0
3. 3x² - 5x + 2 = 0
4. x² + 6x + 9 = 0
5. 4x² - 4x - 3 = 0
Completing the Square
Complete the square for each quadratic expression and solve for x. Show all steps clearly.
1. x² + 6x + 5 = 0
2. 2x² + 8x + 6 = 0
3. x² - 10x + 21 = 0
4. 3x² + 12x + 9 = 0
Graphical Solutions
For each quadratic equation, sketch the graph and identify the roots. Use graph paper if necessary.
1. y = x² - 4x + 3
2. y = 2x² - 3x - 5
3. y = -x² + 6x - 8
Real-World Application
Solve the following problem using quadratic equations. Show all your workings.
A ball is thrown upwards with an initial velocity of 20 m/s from a height of 2 m. The height of the ball above the ground after t seconds is given by the equation h(t) = -5t² + 20t + 2. Determine the time when the ball hits the ground.
Discriminant Analysis
Calculate the discriminant for each quadratic equation and determine the nature of its roots.
1. x² + 2x + 1 = 0
2. 3x² - 4x + 2 = 0
3. x² - 6x + 9 = 0
4. 2x² + 5x + 3 = 0
Factorising Quadratics
Factorise each quadratic expression completely and solve for x.
1. x² - 9x + 20 = 0
2. 2x² - 8x + 6 = 0
3. x² + 5x + 6 = 0
Quadratic Inequalities
Solve the following quadratic inequalities and represent the solution on a number line.
1. x² - 3x < 4
2. 2x² + x > 1
Quadratic Sequences
Determine the nth term of the quadratic sequence and find the 10th term.
1. 2, 6, 12, 20, ...
2. 3, 7, 13, 21, ...
Activity 1: x = 5, x = -1 ; x = 0.5, x = -2 ; x = 1, x = 2/3 ; x = -3 ; x = 1.5, x = -0.5
Activity 2: x = -3 ± √4 ; x = -2 ± √2 ; x = 5 ± √4 ; x = -2 ± √3
Activity 3: Roots at x = 1, x = 3 ; Roots at x = -0.5, x = 2.5 ; Roots at x = 2, x = 4
Activity 4: t ≈ 4.04 seconds
Activity 5: 0 (1 real root) ; 4 (2 real roots) ; 0 (1 real root) ; -11 (no real roots)
Activity 6: (x - 5)(x - 4) ; (2x - 1)(x - 3) ; (x + 2)(x + 3)
Activity 7: x < 4 or x > -1 ; x > -1/2 or x < -1
Activity 8: nth term = n² + n ; nth term = n² + 2n ; 10th term = 110 ; 10th term = 130
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