Syllabus Edition
First teaching 2023
First exams 2025
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Completing the Square (CIE IGCSE Maths: Extended)
Revision Note
Author
MarkExpertise
Maths
Completing the Square
How can I rewrite the first two terms of a quadratic expression as the difference of two squares?
- Look at the quadratic expression x2 + bx + c
- The first two terms can be written as the difference of two squares using the following rule
is the same as where is half of
- Check this is true by expanding the right-hand side
- Is the same as ?
- Yes: (x + 1)(x + 1) - 12 = x2 + 2x + 1 - 1 = x2 + 2x
- Is the same as ?
- This works for negative values of b too
- can be written as which is
- A negative b does not change the sign at the end
How do I complete the square?
- Completing the square is a way to rewrite a quadratic expression in a form containing a squared-bracket
- To complete the square on x2 + 10x + 9
- Use the rule above to replace the first two terms, x2 + 10x, with (x + 5)2 - 52
- then add 9: (x + 5)2 - 52 + 9
- simplify the numbers: (x + 5)2 - 25 + 9
- answer: (x + 5)2 - 16
How do I complete the square when there is a coefficient in front of the x2 term?
- You first need to take out as a factor of the x2 and x terms only
-
- Use square-shaped brackets here to avoid confusion with curly brackets later
-
- Then complete the square on the bit inside the square-brackets:
- This gives
- where p is half of
- This gives
- Finally multiply this expression by the a outside the square-brackets and add the c
- This looks far more complicated than it is in practice!
- Usually you are asked to give your final answer in the form
- For quadratics like , do the above with a = -1
How do I find the turning point by completing the square?
- Completing the square helps us find the turning point on a quadratic graph
- If then the turning point is at
- Notice the negative sign in the x-coordinate
- This links to transformations of graphs (translating by p to the left and q up)
- If then the turning point is still at
- It's at a minimum point if a > 0
- It's at a maximum point if a < 0
- If then the turning point is at
- It can also help you create the equation of a quadratic when given the turning point
- It can also be used to prove and/or show results using the fact that any "squared term", i.e. the bracket (x ± p)2, will always be greater than or equal to 0
- You cannot square a number and get a negative value
Exam Tip
- To know if you have completed the square correctly, expand your answer to check.
Worked example
(a)
By completing the square, find the coordinates of the turning point on the graph of .
Find half of +6 (call this p)
Write x2 + 6x in the form (x + p)2 - p2
is the same as
Put this result into the equation of the curve
Simplify the numbers
Use that the turning point of is at
p = 3 and q = -20
turning point at (-3, -20)
(b)
Write in the form
Factorise -3 out of the first two terms only
Use square-shaped brackets
Complete the square on the x2 - 4x inside the brackets (write in the form (x + p)2 - p2 where p is half of -4)
Simplify the numbers inside the brackets
(-2)2 is 4
Multiply -3 by all the terms inside the square-shaped brackets
Simplify the numbers
This is now in the form a(x + p)2 + q where a = -3, p = -2 and q = 36
Solving by Completing the Square
How do I solve a quadratic equation by completing the square?
- To solve x2 + bx + c = 0
- replace the first two terms, x2 + bx, with (x + p)2 - p2 where p is half of b
- this is called completing the square
- x2 + bx + c = 0 becomes
- (x + p)2 - p2 + c = 0 where p is half of b
- x2 + bx + c = 0 becomes
- rearrange this equation to make x the subject (using ±√)
- For example, solve x2 + 10x + 9 = 0 by completing the square
- x2 + 10x becomes (x + 5)2 - 52
- so x2 + 10x + 9 = 0 becomes (x + 5)2 - 52 + 9 = 0
- make x the subject (using ±√)
- (x + 5)2 - 25 + 9 = 0
- (x + 5)2 = 16
- x + 5 = ±√16
- x = ±4 - 5
- x = -1 or x = -9
- If the equation is ax2 + bx + c = 0 with a number in front of x2, then divide both sides by a first, before completing the square
How does completing the square link to the quadratic formula?
- The quadratic formula actually comes from completing the square to solve ax2 + bx + c = 0
- a, b and c are left as letters, to be as general as possible
- You can see hints of this when you solve quadratics
- For example, solving x2 + 10x + 9 = 0
- by completing the square, (x + 5)2 = 16 so x = ± 4 - 5 (from above)
- by the quadratic formula, = ± 4 - 5 (the same structure)
- For example, solving x2 + 10x + 9 = 0
Exam Tip
- When making x the subject to find the solutions at the end, don't expand the squared brackets back out again!
- Remember to use ±√ to get two solutions
Worked example
Solve by completing the square
Divide both sides by 2 to make the quadratic start with x2
Halve the middle number, -4, to get -2
Replace the first two terms, x2 - 4x, with (x - 2)2 - (-2)2
Simplify the numbers
Add 16 to both sides
Square root both sides
Include the ± sign to get two solutions
Add 2 to both sides
Work out each solution separately
x = 6 or x = -2
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