Completing the Square Video Lesson
How to Complete the Square
To complete the square for ๐ฅ2+b๐ฅ+c:
- Divide b by 2 and write this inside brackets squared as (๐ฅ + b/2)2
- Square the value of b/2 and subtract it.
- Add the constant term of c to this result.
- The final result is given by (๐ฅ + b/2)2 + c โ (b/2)2.
For example, complete the square for y = ๐ฅ2 + 4๐ฅ + 1.
![How to Complete the Square โ mathsathome.com (1) How to Complete the Square โ mathsathome.com (1)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/example-of-how-to-complete-the-square-1024x582.png)
Step 1. Divide the coefficient of ๐ฅ by 2
The coefficient of ๐ฅ is 4.
4 รท 2 = 2 and so, we write (๐ฅ + 2)2.
This step is shown with a purple arrow.
Step 2. Square the value found in step 1 and subtract it
The value found in step 1 is half of the ๐ฅ coefficient. Half of 4 is 2.
We square 2 and subtract this.
22 = 4 and so we subtract 4.
This step is shown with the green arrow.
Step 3. Bring down the constant term
In the equation y = ๐ฅ2 + 4๐ฅ + 1, the constant term is 1.
We add this value to our result.
This step is shown with the red arrow.
Finally, we simplify the constant terms.
-4 + 1 = -3 and so we write this as y = (๐ฅ + 2)2 โ 3.
Formula for Completing the Square
The formula for completing the square for a quadratic, y = ๐ฅ2 + b๐ฅ + c, is .
![How to Complete the Square โ mathsathome.com (3) How to Complete the Square โ mathsathome.com (3)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/complete-the-square-formula-1024x583.png)
If y = ๐ฅ2 + 6๐ฅ + 4, then b = 6 and c = 4.
Therefore b/2 = 3.
Therefore becomes
.
We can simplify 4 โ (3)2 to 4 โ 9 which equals -5.
The final form of the answer is y = (๐ฅ + 3)2 โ 5.
![How to Complete the Square โ mathsathome.com (6) How to Complete the Square โ mathsathome.com (6)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/using-the-formula-to-complete-the-square-1024x581.png)
Here is an example of completing the square when the value of b is odd.
Complete the square for y = ๐ฅ2 + 3๐ฅ + 3.
When the coefficient of ๐ฅ is odd, write it as a fraction over 2 when completing the square.
Step 1. Divide the coefficient of ๐ฅ by 2
The coefficient of ๐ฅ is 3.
To divide 3 by 2, write this as 3/2.
We write this inside the bracket as (๐ฅ + 3/2)2.
This step is written in purple below.
Step 2. Square this value and subtract it
To square 3/2, square the numerator and denominator separately.
32 = 9 and 22 = 4.
We subtract 9/4 from the brackets written in step 1.
This step is written in green below.
Step 3. Bring down the constant term
The constant term of y = ๐ฅ2 + 3๐ฅ + 3 is the number on its own. It is 3.
We add 3 to the equation built so far.
This step is shown in red below.
![How to Complete the Square โ mathsathome.com (7) How to Complete the Square โ mathsathome.com (7)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/completing-the-square-with-an-odd-b-1024x581.png)
Finally, we simplify by collecting the constant terms of โ9/4 and + 3.
3 can be written as 12/4.
โ9/4 + 12/4 = 3/4.
Therefore the result is y = (๐ฅ + 3/2)2 + 3/4.
What is Completing the Square?
Completing the square is an algebraic method used to rearrange a quadratic equation from y = a๐ฅ2+b๐ฅ+c to the form of y = a(๐ฅ+b)2+c. Completing the square allows us to solve quadratic equations that cannot be factorised and to find the turning point of a quadratic. It is possible to complete the square for any quadratic equation.
For example, completing the square with y = ๐ฅ2 + 4๐ฅ + 1 rearranges the equation to y = (๐ฅ + 2)2 โ 3.
y = ๐ฅ2 + 4๐ฅ + 1 and y = (๐ฅ + 2)2 โ 3 are the same equation, just rearranged.
![How to Complete the Square โ mathsathome.com (8) How to Complete the Square โ mathsathome.com (8)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/what-is-completing-the-square-1024x580.png)
Proof of Completing the Square
Completing the square can be proven algebraically by expanding (๐ฅ + b/2)2 to get ๐ฅ2 + b๐ฅ + (b/2)2. This means that any quadratic of the form ๐ฅ2 + b๐ฅ can be rearranged to (๐ฅ + b/2)2 โ (b/2)2.
Adding a constant term of c to both sides of the equation, any quadratic of the form ๐ฅ2 + b๐ฅ + c can be written as .
Visual Proof of Completing the Square
Completing the square can be shown visually using the following steps.
- Consider a square of area ๐ฅ2 and a rectangle of area b๐ฅ.
- Divide the rectangle b๐ฅ in half to form two rectangles of area b/2 ๐ฅ.
- Place these two rectangles around the square of ๐ฅ2
![How to Complete the Square โ mathsathome.com (13) How to Complete the Square โ mathsathome.com (13)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/complete-the-square-proof-1024x580.png)
4. The square of ๐ฅ2 plus the two rectangles almost form a square with side lengths of ๐ฅ + b/2. However, there is a small square of side length b/2 missing.
The area of the overall square (outlined in red above) would be (๐ฅ + b/2)2.
The area of the purple square above would be (b/2)2.
Therefore (๐ฅ + b/2)2 โ (b/2)2 = ๐ฅ2 + b๐ฅ.
Adding a constant term of c to each side of the equation tells us that .
How to Complete the Square with a Coefficient
To complete the square with a coefficient of ๐ฅ2:
- Factorise the coefficient
- Complete the square inside the bracket
- Simplify the terms inside the bracket Expand the bracket
For example, complete the square for y = 2๐ฅ2 + 8๐ฅ โ 6.
Step 1. Factorise the coefficient
The coefficient of ๐ฅ2 is 2.
We factorise the expression by bringing a 2 in front of the brackets and dividing every term inside the brackets by 2.
We get y = 2[๐ฅ2 + 4๐ฅ โ 3]
![How to Complete the Square โ mathsathome.com (15) How to Complete the Square โ mathsathome.com (15)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/how-to-complete-the-square-with-a-coefficient-1024x580.png)
Step 2. Complete the square inside the bracket
Inside the bracket of y = 2[๐ฅ2 + 4๐ฅ โ 3] is ๐ฅ2 + 4๐ฅ โ 3. We complete the square with this.
i) We halve the ๐ฅ coefficient of 4 to get 2. Therefore inside the bracket we have (๐ฅ + 2)2.
ii) We subtract this value squared. 2 squared is 4 so we get (๐ฅ + 2)2 โ 4.
iii) We bring the constant term down. The constant term is -3 so we get (๐ฅ + 2)2 โ 4 โ 3.
Step 3. Simplify the terms inside the bracket
We can simplify (๐ฅ + 2)2 โ 4 โ 3 to (๐ฅ + 2)2 โ 7.
Step 4. Expand the bracket
We have y = 2[ (๐ฅ + 2)2 โ 7 ].
To expand this, we multiply the (๐ฅ + 2)2 term and the -7 term both by 2.
We get y = 2(๐ฅ + 2)2 โ 14.
Completing the Square with a Negative Coefficient
In this example we will complete the square with a negative coefficient of ๐ฅ2.
Complete the square for y = -3๐ฅ2 + 6๐ฅ โ 9. In this example, the coefficient of ๐ฅ2 is -3.
Step 1. Factorise the coefficient of ๐ฅ2
We factorise the coefficient of -3 by writing -3 in front of the brackets and dividing each term within the brackets by -3.
Because we are factorising a negative number, the signs of the terms inside the bracket switch from + to โ or โ to + respectively.
y = -3๐ฅ2 + 6๐ฅ โ 9 becomes y = -3[ ๐ฅ2 โ 2๐ฅ + 3 ].
![How to Complete the Square โ mathsathome.com (16) How to Complete the Square โ mathsathome.com (16)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/completing-the-square-with-a-negative-coefficient-1024x579.png)
Step 2. Complete the square inside the bracket
Inside the bracket of y = -3[๐ฅ2 โ 2๐ฅ + 3] is ๐ฅ2 โ 2๐ฅ + 3. We complete the square with this.
i) We halve the ๐ฅ coefficient of -2 to get -1. Therefore inside the bracket we have (๐ฅ โ 1)2.
ii) We subtract this value squared. (-1) squared is 1 so we get (๐ฅ โ 1)2 โ 1.
iii) We bring the constant term down. The constant term is 3 so we get (๐ฅ โ 1)2 โ 1 + 3.
Step 3. Simplify terms inside the bracket
Inside the bracket, (๐ฅ โ 1)2 โ 1 + 3 can be simplified to (๐ฅ โ 1)2 + 2.
Step 4. Expand the bracket
We have y = -3[ (๐ฅ โ 1)2 + 2 ].
To expand this, we multiply the (๐ฅ โ 1)2 term and the +2 term both by -3.
We get y = -3(๐ฅ โ 1)2 โ 6.
How to Solve an Equation by Completing the Square
Any quadratic equation can be solved by completing the square. After completing the square, move the constant terms so that they are on a different side of the equation to the squared brackets to get (๐ฅ + a)2 = k. Then square root both sides to get ๐ฅ + a = ยฑโk. Subtracting a from both sides gives the solution as ๐ฅ = -a ยฑโk.
For example, solve the equation ๐ฅ2 + 8๐ฅ + 5 = 0.
Step 1. Complete the square
i) Halve the coefficient of ๐ฅ.
The coefficient of ๐ฅ is 8 and half of 8 is 4. We start with writing (๐ฅ + 4)2.
ii) Square this value and subtract it
4 squared is 16, so we subtract 16 to get (๐ฅ + 4)2 โ 16
iii) Bring down the constant term.
The constant term of ๐ฅ2 + 8๐ฅ + 5 is 5, so we add 5 to get (๐ฅ + 4)2 โ 16 + 5.
This simplifies to (๐ฅ + 4)2 โ 11.
![How to Complete the Square โ mathsathome.com (17) How to Complete the Square โ mathsathome.com (17)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/solving-an-equation-by-completing-the-square-1024x579.png)
Step 2. Solve for ๐ฅ
i) First separate the squared brackets and any constant terms on either side of the equals sign
We add 11 to both sides so that (๐ฅ + 4)2 is on the left of the equals sign and 11 is on the right of it.
ii) Square root both sides of the equation
The square root of (๐ฅ + 4)2 is just ๐ฅ + 4.
The square root of 11 is ยฑโ11. We need to remember to take both the positive an negative solutions.
iii) Solve for ๐ฅ
We have ๐ฅ + 4 = ยฑโ11.
We subtract 4 from both sides to get ๐ฅ = -4 ยฑโ11.
Here is another example of solving a quadratic equation by completing the square.
For example, solve the equation ๐ฅ2 + 6๐ฅ โ 1 = 0.
![How to Complete the Square โ mathsathome.com (18) How to Complete the Square โ mathsathome.com (18)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/how-to-solve-an-equation-by-completing-the-square-1024x578.png)
Step 1. Complete the square
i) Halve the coefficient of ๐ฅ.
The coefficient of ๐ฅ is 6 and half of 6 is 3. We start with writing (๐ฅ + 3)2.
ii) Square this value and subtract it
3 squared is 9, so we subtract 9 to get (๐ฅ + 3)2 โ 9
iii) Bring down the constant term.
The constant term of ๐ฅ2 + 6๐ฅ โ 1 is -1, so we subtract 1 to get (๐ฅ + 3)2 โ 10.
Step 2. Solve for ๐ฅ
i) First separate the squared brackets and any constant terms on either side of the equals sign
We add 10 to both sides so that (๐ฅ + 3)2 is on the left of the equals sign and 10 is on the right of it.
ii) Square root both sides of the equation
The square root of (๐ฅ + 3)2 is just ๐ฅ + 3.
The square root of 10 is ยฑโ10. We need to remember to take both the positive an negative solutions.
iii) Solve for ๐ฅ
We have ๐ฅ + 3 = ยฑโ10.
We subtract 3 from both sides to get ๐ฅ = -3 ยฑโ10.
Finding the Vertex by Completing the Square
Completing the square allows a quadratic to be written in the form y = a(๐ฅ + h)2 + k. The vertex of the parabola is found at the coordinate (-h, k). For example, the parabola y = 2(๐ฅ + 3)2 + 1 has a vertex at the coordinate (-3, 1).
The vertex is the turning point of a parabola.
The graph below shows the vertex of y = 2(๐ฅ + 3)2 + 1 at (-3, 1).
![How to Complete the Square โ mathsathome.com (19) How to Complete the Square โ mathsathome.com (19)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/finding-the-vertex-by-completing-the-square-1024x579.png)
The vertex is found at the ๐ฅ value that causes the brackets to be equal to zero.
In the equation y = 2(๐ฅ + 3)2 + 1, the brackets are squared.
When we square a value, the result is always positive. Therefore the bracket squared can never be negative.
This means that the smallest value that can be output by the equation y = 2(๐ฅ + 3)2 + 1 is when the bracket is worth zero.
The value of ๐ฅ that causes this to happen is at ๐ฅ = -3.
When ๐ฅ = -3, the equation of y = 2(๐ฅ + 3)2 + 1 becomes y = 1.
Therefore we find the minimum value at ๐ฅ = -3, y = 1 or written as a coordinate (-3, 1).
Here is an example of finding the vertex of a quadratic by completing the square.
For example, find the vertex of y = ๐ฅ2 โ 2๐ฅ + 3
![How to Complete the Square โ mathsathome.com (20) How to Complete the Square โ mathsathome.com (20)](https://i0.wp.com/mathsathome.com/wp-content/uploads/2021/11/complete-the-square-to-find-the-vertex-1024x578.png)
Step 1. Complete the square
i) Halve the coefficient of ๐ฅ.
The coefficient of ๐ฅ is -2 and half of -2 is -1. We start with writing (๐ฅ โ 1)2.
ii) Square this value and subtract it
(-1) squared is 1, so we subtract 1 to get (๐ฅ โ 1)2 โ 1
iii) Bring down the constant term.
The constant term of ๐ฅ2 โ 2๐ฅ + 3 is 3, so we add 3 to get (๐ฅ โ 1)2 + 2.
Step 2. Read the vertex from the complete the square form
The ๐ฅ coordinate of the vertex is the value of ๐ฅ that makes the bracket equal to zero.
In y = (๐ฅ โ 1)2 + 2, the expression inside the bracket is ๐ฅ โ 1.
This is equal to zero when ๐ฅ = 1. Therefore the ๐ฅ coordinate of the vertex is at ๐ฅ = 1.
The y coordinate of the vertex is simply the constant term in y = (๐ฅ โ 1)2 + 2. This is just the +2 at the end.
The y coordinate of the vertex is at y = 2.
Therefore the coordinate of the vertex is at (-1 , 2).
Notice how the value of the ๐ฅ coordinate is opposite sign to the sign written in the brackets but the y coordinate is the same sign as the constant term at the end.
Here are some examples of reading the turning point from an equation in complete the square form.
Equation | Coordinates of the Vertex |
y = a(๐ฅ + h)2 + k | (-h, k) |
y = (๐ฅ + 3)2 + 5 | (-3, 5) |
y = 5(๐ฅ + 1)2 + 2 | (-1, 2) |
y = (๐ฅ โ 3)2 + 7 | (3, 7) |
y = 2(๐ฅ โ 4)2 โ 3 | (4, -3) |
y = 0.1(๐ฅ + 2)2 โ 6 | (-2, -6) |
y = (๐ฅ โ 5)2 | (5, 0) |