GCSE - General Certificate of Secondary Education Solving Quadratic Equations Questions and Answers — Questions and Answers
Question 1: A rectangular garden has a length that is 5 metres longer than its width. If the area of the garden is 84 m², what is the width of the garden?
- 12 m
- 7 m (Correct answer)
- 14 m
- 5 m
Correct answer: 7 m
Let the width be 'w'. The length is 'w + 5'. The area is width × length, so w(w + 5) = 84. This expands to the quadratic equation w² + 5w - 84 = 0. Factoring this equation gives (w + 12)(w - 7) = 0. The possible solutions are w = -12 or w = 7. Since the width cannot be a negative value, the width must be 7 m.
Question 2: Which of the following are the solutions to the quadratic equation 2x² - 5x - 3 = 0?
- x = -3 and x = 0.5
- x = 3 and x = -2
- x = 3 and x = -0.5 (Correct answer)
- x = 1.5 and x = -1
Correct answer: x = 3 and x = -0.5
The equation 2x² - 5x - 3 = 0 can be solved by factoring. We look for two numbers that multiply to 2*(-3) = -6 and add to -5. These numbers are -6 and +1. We rewrite the middle term: 2x² - 6x + x - 3 = 0. Factor by grouping: 2x(x - 3) + 1(x - 3) = 0. This gives (2x + 1)(x - 3) = 0. Setting each bracket to zero gives 2x + 1 = 0 (so x = -0.5) and x - 3 = 0 (so x = 3).
Question 3: A student is solving the quadratic equation x² + 8x - 7 = 0 by completing the square. Which of the following shows the equation correctly rewritten in the form (x + p)² = q?
- (x + 4)² = 9
- (x + 8)² = 71
- (x - 4)² = 23
- (x + 4)² = 23 (Correct answer)
Correct answer: (x + 4)² = 23
To complete the square for x² + 8x - 7 = 0, first move the constant term to the right side: x² + 8x = 7. Then, take half of the coefficient of x (which is 8/2 = 4), square it (4² = 16), and add it to both sides: x² + 8x + 16 = 7 + 16. The left side is now a perfect square: (x + 4)². The right side is 23. So, the equation is (x + 4)² = 23.
Question 4: Using the quadratic formula, what are the solutions to 3x² + 7x - 1 = 0? Give your answers correct to 2 decimal places.
- x = 0.14 and x = -2.47 (Correct answer)
- x = -0.14 and x = 2.47
- x = 0.15 and x = -2.15
- x = 2.33 and x = -1.00
Correct answer: x = 0.14 and x = -2.47
Using the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a, with a=3, b=7, and c=-1. Substitute the values: x = [-7 ± sqrt(7² - 4*3*(-1))] / (2*3). This simplifies to x = [-7 ± sqrt(49 + 12)] / 6, which is x = [-7 ± sqrt(61)] / 6. The two solutions are x = (-7 + 7.810...) / 6 ≈ 0.14 and x = (-7 - 7.810...) / 6 ≈ -2.47.
Question 5: How many distinct real solutions does the quadratic equation 4x² - 12x + 9 = 0 have?
- Two
- None
- One (Correct answer)
- Infinitely many
Correct answer: One
The number of real solutions can be determined by the discriminant, b² - 4ac. For this equation, a=4, b=-12, and c=9. The discriminant is (-12)² - 4(4)(9) = 144 - 144 = 0. When the discriminant is equal to 0, there is exactly one real solution (also known as a repeated root).
Question 6: Solve the equation x² - 11x + 18 = 0 by factorising.
- x = -2 and x = -9
- x = 2 and x = 9 (Correct answer)
- x = 3 and x = 6
- x = -3 and x = -6
Correct answer: x = 2 and x = 9
To factorise the quadratic x² - 11x + 18 = 0, we need to find two numbers that multiply to +18 and add up to -11. These two numbers are -2 and -9. Therefore, the equation can be written in factored form as (x - 2)(x - 9) = 0. Setting each factor to zero gives the solutions x = 2 and x = 9.
GCSE - General Certificate of Secondary Education Solving Quadratic Equations Questions and Answers
A rectangular garden has a length that is 5 metres longer than its width.
If the area of the garden is 84 m², what is the width of the garden?