Completing the square, and maximum and minimum values Cambridge IGCSE Additional Mathematics revision
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In plain words
Every quadratic has one turning point: the bottom of a U or the top of a hill. Completing the square rewrites the quadratic so that the turning point can be read straight off.
6 things to know
- x² + bx + c = (x + b/2)² − (b/2)² + c. Halve the coefficient of x, square it, and subtract it.
- In the form a(x + p)² + q, the turning point is at (−p, q).
- If a is positive, q is the minimum value. If a is negative, q is the maximum value.
- When the coefficient of x² is not 1, take it out as a factor from the x² and x terms first.
- The turning point gives the range: for a minimum q, the range is f(x) ≥ q.
- The turning point can also be found by differentiating and setting the derivative equal to 0.
Worked example
Write x² − 6x + 11 in the form (x + a)² + b, and state the minimum value.
- Half of −6 is −3. (x − 3)² = x² − 6x + 9.
- So x² − 6x + 11 = (x − 3)² − 9 + 11 = (x − 3)² + 2.
- The minimum value is 2, when x = 3.
Worked example
Write 2x² + 8x + 3 in the form a(x + p)² + q.
- Take out the 2: 2(x² + 4x) + 3.
- Complete the square inside: 2[(x + 2)² − 4] + 3.
- Multiply out the bracket: 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5. The minimum value is −5, when x = −2.
Tips and tricks
- The x-coordinate of the turning point has the opposite sign to the number in the bracket: (x − 3)² gives x = 3.
- Check by expanding your answer. It should give back the original quadratic.
It lands in your notebook with its questions as flashcards.
Completing the square, and maximum and minimum values: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is the minimum value of (x + 4)² − 3?
The square is at least 0, so the smallest value is −3.
What is x² − 8x + 20 in completed square form?
(x − 4)² = x² − 8x + 16, and 20 − 16 = 4.
What are the coordinates of the turning point of y = (x − 1)² + 6?
The bracket is 0 when x = 1.
What is the maximum value of 7 − (x + 2)²?
Something at least 0 is being subtracted, so the most it can be is 7.
What is 3x² − 12x + 5 in completed square form?
3(x − 2)² = 3x² − 12x + 12, and 5 − 12 = −7.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
2 questions, 6 marks. Write your answers on paper, then check them.
Completing the square, and maximum and minimum values
Cambridge IGCSE Additional Mathematics 0606 · 6 marks · papermunch.org
Name ______________________________ Date ______________
Write x² + 10x + 18 in the form (x + a)² + b, and state the minimum value and the value of x at which it occurs.[3]
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(x + 5)² − 7. The minimum value is −7, when x = −5.
Write 5 + 4x − x² in the form a − (x + b)², and state the maximum value.[3]
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9 − (x − 2)². The maximum value is 9, when x = 2.
Answers: Completing the square, and maximum and minimum values
- 1. (x + 5)² − 7. The minimum value is −7, when x = −5.
- 2. 9 − (x − 2)². The maximum value is 9, when x = 2.