Completing the Square Part 2
This is just an extension of being able to complete the square. You need to go through the completing the square part 1 page and the factorising (1 bracket) page before you go through this page and you
There is really nothing new on this page, the questions that we went through in the completing the square part 1 page had quadratic equations in the form of ax2 + bx + c, where a was equal to 1. When a does not equal 1, there is a slight tweak on how we complete the square. It is not hard.
Lets go through a question where a is not 1.
Question 1: Express 2x2 + 12x + 8 in the form m(x + y)2 - d.
The quadratic equation 2x2 + 12x + 8 is in the form ax2 + bx + c.
We can rewrite 2x2 + 12x + 8 as 2(x2 + 6x + 4). We have written the quadratic equation as a product of a (which is 2 in this case) and another quadratic expression. How do we get this other quadratic expression, in this case how did we get (x2 + 6x + 4)? Well we factored out the a (which is 2). You need to understand how to factorise.
So we get 2( ) = 2x2 + 12x + 8
So how do we get the terms inside the bracket?
2x2 = 2 X x2, so x2 needs to be inside the brackets
We have used the symbol X to represent the multiplication sign here, so you don't confuse the term x with the multiplication sign.
2 multiplied by something needs to equal 12x, lets represent that something using the letter h.So if 2h = 12x, then h = 6x. Thus 6x needs to be in the bracket.
2 multiplied by something needs to equal 8, lets represent that something using the letter i.So if 2i = 8, i = 4. Thus 4 needs to be in the bracket.
So now we have 2x2 + 12x + 8 = 2(x2 + 6x + 4)
We can write now (x2 + 6x + 4) in the form of (x + y)2 - e.
We know how to complete the square, as learned in the completing the square part 1 page.
We can first focus on attaining the (x2 + 6x) part of the quadratic equation.
If we expand (x + y)2 we get x2 + 2yx + y2 .
We see that we attain the x2 term. The coefficient of the x term is 2y, we need it to be 6, so we can solve to find y.
2y = 6
(Divide both sides by 2)
2y/2 = 6/2
y = 3
So now we have (x + 3)2, as y = 3. When we expand (x + 3)2 we get the (x2 + 6x) part of the quadratic expression, however we also get + 9, as shown below.
(x + 3)2 = x2 + 6x + 9
We need only x2 + 6x
So we can subtract 9 from both sides, as shown below (remember whatever we do to one side, we must do to the other).
(x + 3)2 - 9 = x2 + 6x + 9 - 9
(x + 3)2 - 9 = x2 + 6x
We can now add 4 to both sides. Why? Well we need x2 + 6x + 4, as that is the quadratic expression which we need to write in the form (x + y)2 - e.
(x + 3)2 - 9 + 4 = x2 + 6x + 4
(x + 3)2 - 5 = x2 + 6x + 4
Thus y = 3 and e = 5
Remember that the question we are trying to solve is to express 2x2 + 12x + 8 in the form m(x + y)2 - d.
So if
2x2 + 12x + 8 = 2(x2 + 6x + 4)
and
x2 + 6x + 4 = (x + 3)2 - 5, then we can write
2(x2 + 6x + 4) = 2((x + 3)2 - 5) (Read this carefully and look at the brackets)
We can expand 2((x + 3)2 - 5), as shown below.
2((x + 3)2 - 5) = 2(x + 3)2 - 10
Thus we have expressed 2x2 + 12x + 8 in the form m(x + y)2 - d, where m = 2, y = 3 and d = 10.
One thing to understand is that when we need to express a quadratic equation ax2 + bx + c in the form m(x + y)2 - d, we first write the quadratic equation as a product of a and another quadratic expression. We do this by factoring out the a. So we get
a( ) = ax2 + bx + c
If we had a question where we needed to express -5x2 + 7x - 9 in the form m(x + y)2 - d, then we would factor out -5, so we can first express
-5x2 + 7x - 9 as -5( )
So what do we put inside the brackets, so that when we expand -5( ) we get -5x2 + 7x - 9.
So -5 multiplied by something is equal to -5x2, lets represent that something with the letter h.So -5h = -5x2, therefore h = x2
How did we get h? We divided both sides by -5.
So we can write -5(x2 )
-5 multiplied by something is equal to 7x, lets represent that something with the letter i.So -5i = 7x, therefore i = 7x/-5
How did we get i? We divided both sides by -5.
So we can write -5(x2 - 7x/5 )
-5 multiplied by something is equal to -9, lets represent that something with the letter j.So -5j = -9, therefore j = -9/-5 = 9/5
How did we get j? We divided both sides by -5.
So we can write -5(x2 - 7x/5 + 9/5)
Now we can write (x2 - 7x/5 + 9/5) in the form (x + y)2 - e.
We know how to complete the square, as learned in the completing the square part 1 page.
Lets first focus on attaining the (x2 - 7x/5) part of the equation.
If we expand (x + y)2 we get x2 + 2yx + y2
We see that we attain the x2 term. The coefficient of the x term is 2y, we need it to be -7/5, so we can solve to find y.
2y = -7/5
(Divide both sides by 2)
2y/2 = -7/5 ÷ 2
y = -7/5 x 1/2
y = -7/10
So now we have (x - 7/10)2, as y = -7/10. When we expand (x - 7/10)2 we get the (x2 - 7x/5) part of the quadratic expression, however we also get + 49/100 , as shown below.
(x - 7/10)2 = x2 - 7x/5 + 49/100
We need only x2 - 7x/5
So we can subtract 49/100 from both sides, as shown below (remember whatever we do to one side, we must do to the other).
(x - 7/10)2 - 49/100 = x2 - 7x/5 + 49/100 - 49/100
(x - 7/10)2 - 49/100 = x2 - 7x/5
We can now add 9/5 to both sides. Why? Well we need x2 - 7x/5 + 9/5, as that is the quadratic expression which we need to write in the form (x + y)2 - e.
(x - 7/10)2 - 49/100 + 9/5 = x2 + 7x/5 + 9/5
(x - 7/10)2 + 131/100 = x2 + 7x/5 + 9/5
How did we get 131/100?
-49/100 + 9/5 = -49/100 + 180/100 = 131/100
Thus y = -7/10 and e = -131/100
Lets substitute y and e with their respective values into (x + y)2 - e.
(x + -7/10)2 - -131/100 = (x - 7/10)2 + 131/100
So (x - 7/10)2 + 131/100 = x2 + 7x/5 + 9/5
Remember that the question we are trying to solve is to express -5x2 + 7x - 9 in the form m(x + y)2 - d.
So if
-5x2 + 7x - 9 = -5(x2 - 7x/5 + 9/5)
and
x2 - 7x/5 + 9/5 = (x - 7/10)2 + 131/100, then we can write
-5(x2 - 7x/5 + 9/5) = -5((x - 7/10)2 + 131/100) (Read this carefully and look at the brackets)
We can expand -5((x - 7/10)2 + 131/100), as shown below.
-5((x - 7/10)2 + 131/100) = -5(x - 7/10)2 - 655/100
The fraction -655/100 can be simplified, as we divide the numerator and denominator by 5. This gives us -131/20.
So
-5(x - 7/10)2 - 655/100 = -5(x - 7/10)2 - 131/20
Thus we have expressed -5x2 + 7x - 9 in the form m(x + y)2 - d, where m = -5, y = -7/10 and d = 131/20.
There is really nothing new on this page, the questions that we went through in the completing the square part 1 page had quadratic equations in the form of ax2 + bx + c, where a was equal to 1. When a does not equal 1, there is a slight tweak on how we complete the square. It is not hard.
Lets go through a question where a is not 1.
Question 1: Express 2x2 + 12x + 8 in the form m(x + y)2 - d.
The quadratic equation 2x2 + 12x + 8 is in the form ax2 + bx + c.
We can rewrite 2x2 + 12x + 8 as 2(x2 + 6x + 4). We have written the quadratic equation as a product of a (which is 2 in this case) and another quadratic expression. How do we get this other quadratic expression, in this case how did we get (x2 + 6x + 4)? Well we factored out the a (which is 2). You need to understand how to factorise.
So we get 2( ) = 2x2 + 12x + 8
So how do we get the terms inside the bracket?
2x2 = 2 X x2, so x2 needs to be inside the brackets
We have used the symbol X to represent the multiplication sign here, so you don't confuse the term x with the multiplication sign.
2 multiplied by something needs to equal 12x, lets represent that something using the letter h.So if 2h = 12x, then h = 6x. Thus 6x needs to be in the bracket.
2 multiplied by something needs to equal 8, lets represent that something using the letter i.So if 2i = 8, i = 4. Thus 4 needs to be in the bracket.
So now we have 2x2 + 12x + 8 = 2(x2 + 6x + 4)
We can write now (x2 + 6x + 4) in the form of (x + y)2 - e.
We know how to complete the square, as learned in the completing the square part 1 page.
We can first focus on attaining the (x2 + 6x) part of the quadratic equation.
If we expand (x + y)2 we get x2 + 2yx + y2 .
We see that we attain the x2 term. The coefficient of the x term is 2y, we need it to be 6, so we can solve to find y.
2y = 6
(Divide both sides by 2)
2y/2 = 6/2
y = 3
So now we have (x + 3)2, as y = 3. When we expand (x + 3)2 we get the (x2 + 6x) part of the quadratic expression, however we also get + 9, as shown below.
(x + 3)2 = x2 + 6x + 9
We need only x2 + 6x
So we can subtract 9 from both sides, as shown below (remember whatever we do to one side, we must do to the other).
(x + 3)2 - 9 = x2 + 6x + 9 - 9
(x + 3)2 - 9 = x2 + 6x
We can now add 4 to both sides. Why? Well we need x2 + 6x + 4, as that is the quadratic expression which we need to write in the form (x + y)2 - e.
(x + 3)2 - 9 + 4 = x2 + 6x + 4
(x + 3)2 - 5 = x2 + 6x + 4
Thus y = 3 and e = 5
Remember that the question we are trying to solve is to express 2x2 + 12x + 8 in the form m(x + y)2 - d.
So if
2x2 + 12x + 8 = 2(x2 + 6x + 4)
and
x2 + 6x + 4 = (x + 3)2 - 5, then we can write
2(x2 + 6x + 4) = 2((x + 3)2 - 5) (Read this carefully and look at the brackets)
We can expand 2((x + 3)2 - 5), as shown below.
2((x + 3)2 - 5) = 2(x + 3)2 - 10
Thus we have expressed 2x2 + 12x + 8 in the form m(x + y)2 - d, where m = 2, y = 3 and d = 10.
One thing to understand is that when we need to express a quadratic equation ax2 + bx + c in the form m(x + y)2 - d, we first write the quadratic equation as a product of a and another quadratic expression. We do this by factoring out the a. So we get
a( ) = ax2 + bx + c
Another question:
Another question:
If we had a question where we needed to express -5x2 + 7x - 9 in the form m(x + y)2 - d, then we would factor out -5, so we can first express
-5x2 + 7x - 9 as -5( )
So what do we put inside the brackets, so that when we expand -5( ) we get -5x2 + 7x - 9.
So -5 multiplied by something is equal to -5x2, lets represent that something with the letter h.So -5h = -5x2, therefore h = x2
How did we get h? We divided both sides by -5.
So we can write -5(x2 )
-5 multiplied by something is equal to 7x, lets represent that something with the letter i.So -5i = 7x, therefore i = 7x/-5
How did we get i? We divided both sides by -5.
So we can write -5(x2 - 7x/5 )
-5 multiplied by something is equal to -9, lets represent that something with the letter j.So -5j = -9, therefore j = -9/-5 = 9/5
How did we get j? We divided both sides by -5.
So we can write -5(x2 - 7x/5 + 9/5)
Now we can write (x2 - 7x/5 + 9/5) in the form (x + y)2 - e.
We know how to complete the square, as learned in the completing the square part 1 page.
Lets first focus on attaining the (x2 - 7x/5) part of the equation.
If we expand (x + y)2 we get x2 + 2yx + y2
We see that we attain the x2 term. The coefficient of the x term is 2y, we need it to be -7/5, so we can solve to find y.
2y = -7/5
(Divide both sides by 2)
2y/2 = -7/5 ÷ 2
y = -7/5 x 1/2
y = -7/10
So now we have (x - 7/10)2, as y = -7/10. When we expand (x - 7/10)2 we get the (x2 - 7x/5) part of the quadratic expression, however we also get + 49/100 , as shown below.
(x - 7/10)2 = x2 - 7x/5 + 49/100
We need only x2 - 7x/5
So we can subtract 49/100 from both sides, as shown below (remember whatever we do to one side, we must do to the other).
(x - 7/10)2 - 49/100 = x2 - 7x/5 + 49/100 - 49/100
(x - 7/10)2 - 49/100 = x2 - 7x/5
We can now add 9/5 to both sides. Why? Well we need x2 - 7x/5 + 9/5, as that is the quadratic expression which we need to write in the form (x + y)2 - e.
(x - 7/10)2 - 49/100 + 9/5 = x2 + 7x/5 + 9/5
(x - 7/10)2 + 131/100 = x2 + 7x/5 + 9/5
How did we get 131/100?
-49/100 + 9/5 = -49/100 + 180/100 = 131/100
Thus y = -7/10 and e = -131/100
Lets substitute y and e with their respective values into (x + y)2 - e.
(x + -7/10)2 - -131/100 = (x - 7/10)2 + 131/100
So (x - 7/10)2 + 131/100 = x2 + 7x/5 + 9/5
Remember that the question we are trying to solve is to express -5x2 + 7x - 9 in the form m(x + y)2 - d.
So if
-5x2 + 7x - 9 = -5(x2 - 7x/5 + 9/5)
and
x2 - 7x/5 + 9/5 = (x - 7/10)2 + 131/100, then we can write
-5(x2 - 7x/5 + 9/5) = -5((x - 7/10)2 + 131/100) (Read this carefully and look at the brackets)
We can expand -5((x - 7/10)2 + 131/100), as shown below.
-5((x - 7/10)2 + 131/100) = -5(x - 7/10)2 - 655/100
The fraction -655/100 can be simplified, as we divide the numerator and denominator by 5. This gives us -131/20.
So
-5(x - 7/10)2 - 655/100 = -5(x - 7/10)2 - 131/20
Thus we have expressed -5x2 + 7x - 9 in the form m(x + y)2 - d, where m = -5, y = -7/10 and d = 131/20.