Completing the square
AQA GCSE Maths revision on Completing the square. Aligned to the AQA GCSE Mathematics 8300 specification. This bank has 16 practice questions on this topic.
Sample questions (3 of 16)
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Question 1
When expressing x squared + 8x + 3 in the form (x + a) squared + b, what is the geometric purpose of the term (x + a) squared?
- A) To find the x-intercepts
- B) To represent the area of a square with side length x + a
- C) To calculate the gradient of the curve
- D) To eliminate the constant term
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Answer: To represent the area of a square with side length x + a
The method is named completing the square because it physically constructs a square of dimensions (x + a) by (x + a). The expression (x + a) squared represents this geometric square, while the remaining constant term accounts for the area that needs to be added or subtracted to match the original expression.
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Question 2
A student attempts to complete the square for x squared + 6x + 5 by writing (x + 6) squared - 31. Identify the student's fundamental error.
- A) They forgot to square the 6
- B) They used the full coefficient of x instead of halving it
- C) They added the constant instead of subtracting it
- D) They failed to include the x-squared term
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Answer: They used the full coefficient of x instead of halving it
When expanding (x + a) squared, the middle term is 2ax. To match the original coefficient of x, the value inside the bracket must be half of the original coefficient. The student used the full coefficient, which creates a middle term of 12x instead of 6x.
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Question 3
Why is the vertex form of a quadratic, y = a(x - h) squared + k, particularly useful for sketching graphs?
- A) It identifies the y-intercept instantly
- B) It shows the roots of the equation
- C) It reveals the coordinates of the turning point
- D) It calculates the area under the curve
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Answer: It reveals the coordinates of the turning point
In the form a(x - h) squared + k, the value 'h' represents the horizontal shift and 'k' represents the vertical shift from the origin. Because the squared term is always non-negative, the turning point occurs exactly when (x - h) equals zero, making the coordinates (h, k) clear by inspection.
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13 more questions on Completing the square — plus mistakes tracking and spaced repetition across the whole Maths spec.