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When we have quadratic expressions, we can rewrite the expressions in different forms. One way is to factorise the expression, and thus you would attain the quadratic expression as a product of two brackets. To understand this page, please make sure you know how to factorise with 1 bracket; you can check this out on the factorising 1 bracket page.
Completing the square is just another way of writing a quadratic expression; the usefulness of this is explained in the plotting quadratic equations page.
By completing the square we rewrite the quadratic expression as (x - y)2 - d.
Question: Write x2 - 6x + 1 in the form (x -a)2 - b.
We have to rewrite the quadratic expression in the form (x -a)2 - b; we need to find the values of a and b.
Once we attain the values of a and b, we would be able to write x2 - 6x + 1 in the form of (x -a)2 - b.
What does (x -a)2 - b equal? (x -a)2 - b = x2 - 2ax + a2 - b
So we can see that there is already a x2 term.
So we know that to achieve the (x2 - 6x) part of the quadratic expression, it would come from the expansion of (x -a)2 .
(x - a)(x - a) = x2 - 2ax + a2
So this means that the x2 term is achieved already. The coefficient of x is (-2a), we need the coefficient of x to be 6, so-2a equals -6, so we can solve to find a.
-2a = -6
(Divide both sides by -2)
-2a/-2 = -6/-2
a = 3
So this means we get (x - 3)2 = (x - 3)(x - 3) = x2 - 6x + 9 (we achieve the x2 - 6x, however we also get the number 9)
So we can write (x - 3)2 - 9 = x2 - 6x + 9 - 9 = x2 - 6x
Thus (x - 3)2 - 9 = x2 - 6x
We need x2 - 6x + 1 though.
So
(Add 1 to both sides, remember we have to treat both sides equally)
(x - 3)2 - 9 + 1 = x2 - 6x + 1
(x - 3)2 - 8 = x2 - 6x + 1
Therefore x2 - 6x + 1 in the form of (x - a)2 - b is (x - 3)2 - 8, where a = 3, b = 8.
Expand and simplify (x - 3)2 - 8 and see if you get x2 - 6x + 1.
Question 2: Express x2 + 8x + 4 in the form (x + y)2 - d.
(x + y)2 - d = (x + y)(x + y) - d = x2 + xy + xy + y2 - d = x2 + 2xy + y2 - d
Lets first focus on the (x2 + 8x) part of the quadratic expression.
We can see that to achieve the (x2 + 8x) part of the quadratic expression, it would be achieved when (x - y)2 is expanded.
(x + y)2 = x2 + xy + xy + y2 = x2 + 2yx + y2
So we have already achieved the x2 term. The coefficient of the x term is 2y; we need the coefficient of x to be 8.
So lets solve to find the value of y
2y = 8
(Divide both sides by 2)
2y/2 = 8/2
y = 4
So now we can write (x + 4)2 - d = x2 + 8x + 4 . We still need to find d though.
(x + 4)2 = (x + 4)(x + 4) = x2 + 8x + 16
We need only x2 + 8x, we do not need the + 16.
So we can subtract 16 from both sides.
(x + 4)2 = x2 + 8x + 16
(x + 4)2 -16 = x2 + 8x + 16 - 16
(x + 4)2 - 16 = x2 + 8x
We need x2 + 8x + 4 though.
We know that (x + 4)2 - 16 = x2 + 8x, so we can add 4 to both sides
(x + 4)2 - 16 + 4 = x2 + 8x + 4
(x + 4)2 - 12 = x2 + 8x + 4
So x2 + 8x + 4 in the form of (x + y)2 - d is (x + 4)2 - 12, where y = 4 and d = 12.
Expand and simplify (x + 4)2 - 12 and see if you get x2 + 8x + 4.
You need to understand that if we have a quadratic expression x2 + bx + c and we need to express the quadratic expression in the form (x + y)2 - d, you can see that we can attain the ax2 + bx part of the quadratic expression from the (x + y)2.
When we expand (x + y)2 , as shown below, we are able to achieve the (x2 + bx) part of the quadratic expression, given we find a value for y.
We find a value for y, by dividing b by 2. We did this for both questions above.
(x + y)2 = x2 + 2yx + y2
The coefficient of x is 2y, we need the coefficient of x to be b. So we need to find the value of y, which makes 2y equal b.
Solve to find the value of y.
2y = b
(Divide both sides 2)
2y/2 = b/2
y = b/2
Given we now have (x + b/2)2, as y = b/2, (x + b/2)2 = (x + b/2)(x + b/2) = x2 + (b/2)x + (b/2)x + (b/2)2= x2 + bx + (b/2)2
We get the (x2 + bx) part of the quadratic expression which we need but we do not need the + (b/2)2 .
So if (x + b/2)2 = x2 + bx + (b/2)2, we can subtract (b/2)2 from both sides.
(x + b/2)2 - (b/2)2 = x2 + bx + (b/2)2 - (b/2)2
(x + b/2)2 - (b/2)2 = x2 + bx
However, we need to attain the quadratic expression x2 + bx + c.
We can add c to both sides
(x + b/2)2 - (b/2)2 + c = x2 + bx + c
So this tells us, that to express a quadratic expression x2 + bx + c in the form (x + y)2 - d, we can simply substitute the values of b and c into (x + b/2)2 - (b/2)2 + c.
Lets try this with the two below quadratic expressions,
Completing the square is just another way of writing a quadratic expression; the usefulness of this is explained in the plotting quadratic equations page.
By completing the square we rewrite the quadratic expression as (x - y)2 - d.
Video 1 (Explanation)
Video 1 (Explanation)
Question: Write x2 - 6x + 1 in the form (x -a)2 - b.
We have to rewrite the quadratic expression in the form (x -a)2 - b; we need to find the values of a and b.
Once we attain the values of a and b, we would be able to write x2 - 6x + 1 in the form of (x -a)2 - b.
What does (x -a)2 - b equal? (x -a)2 - b = x2 - 2ax + a2 - b
So we can see that there is already a x2 term.
So we know that to achieve the (x2 - 6x) part of the quadratic expression, it would come from the expansion of (x -a)2 .
(x - a)(x - a) = x2 - 2ax + a2
So this means that the x2 term is achieved already. The coefficient of x is (-2a), we need the coefficient of x to be 6, so-2a equals -6, so we can solve to find a.
-2a = -6
(Divide both sides by -2)
-2a/-2 = -6/-2
a = 3
So this means we get (x - 3)2 = (x - 3)(x - 3) = x2 - 6x + 9 (we achieve the x2 - 6x, however we also get the number 9)
So we can write (x - 3)2 - 9 = x2 - 6x + 9 - 9 = x2 - 6x
Thus (x - 3)2 - 9 = x2 - 6x
We need x2 - 6x + 1 though.
So
(Add 1 to both sides, remember we have to treat both sides equally)
(x - 3)2 - 9 + 1 = x2 - 6x + 1
(x - 3)2 - 8 = x2 - 6x + 1
Therefore x2 - 6x + 1 in the form of (x - a)2 - b is (x - 3)2 - 8, where a = 3, b = 8.
Expand and simplify (x - 3)2 - 8 and see if you get x2 - 6x + 1.
Video 2 (Explanation)
Video 2 (Explanation)
Question 2: Express x2 + 8x + 4 in the form (x + y)2 - d.
(x + y)2 - d = (x + y)(x + y) - d = x2 + xy + xy + y2 - d = x2 + 2xy + y2 - d
Lets first focus on the (x2 + 8x) part of the quadratic expression.
We can see that to achieve the (x2 + 8x) part of the quadratic expression, it would be achieved when (x - y)2 is expanded.
(x + y)2 = x2 + xy + xy + y2 = x2 + 2yx + y2
So we have already achieved the x2 term. The coefficient of the x term is 2y; we need the coefficient of x to be 8.
So lets solve to find the value of y
2y = 8
(Divide both sides by 2)
2y/2 = 8/2
y = 4
So now we can write (x + 4)2 - d = x2 + 8x + 4 . We still need to find d though.
(x + 4)2 = (x + 4)(x + 4) = x2 + 8x + 16
We need only x2 + 8x, we do not need the + 16.
So we can subtract 16 from both sides.
(x + 4)2 = x2 + 8x + 16
(x + 4)2 -16 = x2 + 8x + 16 - 16
(x + 4)2 - 16 = x2 + 8x
We need x2 + 8x + 4 though.
We know that (x + 4)2 - 16 = x2 + 8x, so we can add 4 to both sides
(x + 4)2 - 16 + 4 = x2 + 8x + 4
(x + 4)2 - 12 = x2 + 8x + 4
So x2 + 8x + 4 in the form of (x + y)2 - d is (x + 4)2 - 12, where y = 4 and d = 12.
Expand and simplify (x + 4)2 - 12 and see if you get x2 + 8x + 4.
You need to understand that if we have a quadratic expression x2 + bx + c and we need to express the quadratic expression in the form (x + y)2 - d, you can see that we can attain the ax2 + bx part of the quadratic expression from the (x + y)2.
When we expand (x + y)2 , as shown below, we are able to achieve the (x2 + bx) part of the quadratic expression, given we find a value for y.
We find a value for y, by dividing b by 2. We did this for both questions above.
(x + y)2 = x2 + 2yx + y2
The coefficient of x is 2y, we need the coefficient of x to be b. So we need to find the value of y, which makes 2y equal b.
Solve to find the value of y.
2y = b
(Divide both sides 2)
2y/2 = b/2
y = b/2
Given we now have (x + b/2)2, as y = b/2, (x + b/2)2 = (x + b/2)(x + b/2) = x2 + (b/2)x + (b/2)x + (b/2)2= x2 + bx + (b/2)2
We get the (x2 + bx) part of the quadratic expression which we need but we do not need the + (b/2)2 .
So if (x + b/2)2 = x2 + bx + (b/2)2, we can subtract (b/2)2 from both sides.
(x + b/2)2 - (b/2)2 = x2 + bx + (b/2)2 - (b/2)2
(x + b/2)2 - (b/2)2 = x2 + bx
However, we need to attain the quadratic expression x2 + bx + c.
We can add c to both sides
(x + b/2)2 - (b/2)2 + c = x2 + bx + c
So this tells us, that to express a quadratic expression x2 + bx + c in the form (x + y)2 - d, we can simply substitute the values of b and c into (x + b/2)2 - (b/2)2 + c.
Lets try this with the two below quadratic expressions,
- x2 - 6x + 1, here b = -6 and c = 1
- x2 + 8x + 4, here b = 8, c = 4
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