Question 1 Easy
Find the minimum value of x^2+2x+6.
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Answer: 5
- (x+1)^2+5.
- A square cannot be negative.
- Method (1): (x+1)^2+5.
- Accuracy (1): Answer 5
Completing the square practice questions with answers
A second set of completing the square questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.
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Question 2 Easy
Find the minimum value of 3x^2+6x+5.
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Answer: 2
- Factor 3: 3(x^2+2x)+5 = 3[(x+1)^2-1]+5.
- 3(x+1)^2+2. A square cannot be negative, so the least value is 2.
- Method (1): Factor 3: 3(x^2+2x)+5 = 3[(x+1)^2-1]+5.
- Accuracy (1): Answer 2
Question 3 Medium
y=2x^2+12x+11 equals 2(x+p)^2+q. Give p and q as the pair p, q.
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Answer: 3, -7
- Inside: half of 6 is 3.
- 2(x+3)^2-18+11.
- Method (1): Inside: half of 6 is 3.
- Accuracy (1): Answer 3, -7
Question 4 Medium
Write 4x^2-16x+9 in the form a(x+p)^2+q.
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Answer: 4(x-2)^2-7
- 4(x^2-4x)+9.
- 4[(x-2)^2-4]+9.
- Method (1): 4(x^2-4x)+9.
- Accuracy (1): Answer 4(x-2)^2-7
Question 5 Hard
A drone height is h=20+3t-t^2. State the greatest height as a fraction.
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Answer: 89/4
- -(t^2-3t)+20.
- Half of -3 is -3/2, and (3/2)^2=9/4.
- 20+9/4=89/4.
- Method (1): -(t^2-3t)+20.
- Accuracy (1): Answer 89/4
Question 6 Hard
Find the least value of x^2-6x+10 and the x that gives it. Give the x value first, then the least value.
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Answer: 3 then 1
- (x-3)^2+1.
- Least value 1 when x=3.
- Method (1): (x-3)^2+1.
- Accuracy (1): Answer 3 then 1
Exam-style question
A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.
Exam-style question 14 marks
Write 3x^2+6x+1 in completed-square form and give the turning point.
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Answer: 3(x+1)^2-2 with turning point (-1, -2)
- 3(x^2+2x)+1.
- 3[(x+1)^2-1]+1 = 3(x+1)^2-2.
- Turning point (-1, -2).
- Method (1): Factor 3
- Method (1): Complete the square inside
- Accuracy (1): Completed form
- Accuracy (1): Turning point
Quick checks
Short one-step questions. Pick an option before you open the answer.
Check 1
Why do you subtract (b/2)^2 after writing the bracket?
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Because expanding the bracket creates that square, and the original expression did not have it. Because expanding the bracket creates that square, and the original expression did not have it.
Check 2
The graph of y=(x+5)^2-22 turns at which point?
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(-5, -22). The sign inside the bracket flips for the x-coordinate. (-5, -22). The sign inside the bracket flips for the x-coordinate.
Check 3
What does a positive a tell you about the turning point?
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The turning point is a minimum. The curve opens upwards. The turning point is a minimum. The curve opens upwards.
Check 4
What does a negative a tell you?
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The turning point is a maximum. The curve opens downwards. The turning point is a maximum. The curve opens downwards.
Check 5
Why factor a before completing the square?
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The half-and-square step is for an expression that starts with x^2, not with 2x^2 or 3x^2. The half-and-square step is for an expression that starts with x^2, not with 2x^2 or 3x^2.
Check 6
In 2(x+2)^2-3, which number is the least value?
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-3. The squared bracket is never negative. -3. The squared bracket is never negative.
Check 7
How many roots does a(x+p)^2+q=0 have when q and a have opposite signs?
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Two. The square can equal a positive number. Two. The square can equal a positive number.
Check 8
Why is (2x+2)^2 not the completed form of 2x^2+8x?
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Expanding (2x+2)^2 gives 4x^2, so the x^2 term has changed. Expanding (2x+2)^2 gives 4x^2, so the x^2 term has changed.
Check 9
A question says leave the answer in surd form. What should you not do?
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Do not replace the square root with a rounded decimal. Do not replace the square root with a rounded decimal.
Check 10
The point (-2, -3) is the turning point of which completed form with a=2?
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2(x+2)^2-3. x+2 is zero at x=-2. 2(x+2)^2-3. x+2 is zero at x=-2.
Check 11
What is half of the linear coefficient in x^2-4x?
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-2. The bracket is (x-2). -2. The bracket is (x-2).
Check 12
You have 3[(x-2)^2-4]+7. What is the next line?
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3(x-2)^2-12+7, which is 3(x-2)^2-5. Multiply the -4 by 3. 3(x-2)^2-12+7, which is 3(x-2)^2-5. Multiply the -4 by 3.
Check 13
Why can a minimum be negative?
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The constant q can be negative even though the square itself is not. The constant q can be negative even though the square itself is not.
Check 14
Solving (x+1)^2=9 gives which two linear steps?
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x+1=3 or x+1=-3. Both square roots are used. x+1=3 or x+1=-3. Both square roots are used.
Check 15
How do you spot a maximum from 9-(x+2)^2?
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Subtracting a square means 9 is the greatest value the expression can take. Subtracting a square means 9 is the greatest value the expression can take.
Keep going with Completing the square
Practise free
AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.