A quilt is three feet by six feet.
If there is 22 square feet of fabric available
to use as a border around the quilt,
how wide should the border be?
Assume all of the fabric must be used.
Referring to the model below,
the red rectangle represents the quilt,
the blue region represents the border around the quilt.
So because the quilt is three feet by six feet,
this length is three feet,
and this length is six feet.
And now we don't know the width of the border,
but it is of uniform width.
Let's have X equal the width of the border,
which means this length is X feet,
and so is this length, at each corner.
And now as we're going to terming the dimensions
of the outer rectangle, or the quilt,
including the border.
If this length is three, and this length is X,
and this length is X,
this outer length must be three, plus X, plus X,
or three, plus two X feet.
This length here must be six, plus X, plus X,
or six, plus two X feet.
From here, we're going to term in the area of the border,
but a terming the area of the outer rectangle,
and subtracting the area of the inner rectangle.
And this area must be equal to the 22 square feet of fabric
that is used for the border.
And because the area of a rectangle's
equal to length times width,
the area of the outer rectangle,
or the area of the quilt and the border,
must be equal to the quantity
six plus two X, times the quantity three plus two X.
So this gives us the area of the quilt and the border,
or the area of the outer rectangle.
And now we want to subtract the area of the quilt,
or the area of the inner rectangle,
which would be three times six.
This leaves us with the area of the border,
which must equal 22 square feet.
Therefore, this is equal to 22.
Now if we solve this equation for X,
we can determine the width of the border.
For the first step, let's (mumbles) the parenthesis
on the left side.
When multiplying two binomials,
we always have four products.
We first distribute the six,
then distribute the two X.
Six times three equals 18.
Six times two X equals 12 X.
So we have plus 12 X.
And now we distribute two X.
Two X times three equals six X,
which gives us plus six X.
And two X times two X equals four X squared,
Which gives us plus four X squared.
And we have minus three times six,
which gives us minus 18 equals 22.
And now we simplify the left side of the equation.
18 minus 18 is equal to zero.
We have four X squared.
12 X plus six X equals 18 X.
So plus 18 X, equals 22.
Notice how we have a quadratic equation.
Let's set this equal to zero, and solve.
To set the equation equal to zero,
we subtract 22 on both sides.
Simplifying, we have four X squared,
plus 18 X, minus 22, equals zero.
There are several methods to solve a quadratic equation.
For this example, while this is factorable,
let's solve this using the quadratic formula.
Let's work on this on the next slide.
So we could apply the quadratic formula in this form,
where A equals four, B equals 18,
and C equals negative 22.
But instead, let's first factor out
the greatest common factor of two.
The factor of two out of the left side,
we're left with two times the quantity two X squared,
plus nine X, minus 11, equals zero.
In this form, this equation equals zero
when the quadratic inside the parenthesis, is equal to zero.
Which means in this form, we can apply the quadratic formula
where A equals two, B equals nine,
and C equals negative 11.
However if we want to, we can also divide
both sides of the equation by two.
Notice by doing this on the left,
two divided by two equals one.
The left side simplifies to two X squared,
plus nine X, minus 11.
On the right side, zero divided by two is still zero.
So now applying the quadratic formula in this form,
we can use A equals two, B equals nine,
and c equals negative 11.
Let's go ahead and perform this substitution.
Performing this substitution,
we have X equals negative B, is negative nine.
And then we have plus or minus the square root
of B squared, which is the square of nine, or nine squared,
minus four times A times C,
is minus four times two times negative 11.
All this is divided by two times A,
which is two times two.
And now we begin simplifying.
We have X equals, in the numerator, negative nine,
plus or minus the square root
of the square of nine is 81.
And we have minus four times two, times negative 11.
Four times two times negative 11 equals negative 88.
Minus negative 88 is equivalent to plus 88.
We can also think of this as negative four,
times two, times negative 11,
which equals positive 88, and therefore we have plus 88.
Denominator is two times two, which equals four.
Next, 81 plus 88 equals 169.
In the numerator we have negative nine
plus or minus the square root of 169,
all divided by four.
169 is equal to 13 squared, and therefore,
the square root of 169, simplifies perfectly to 13,
which gives us X equals the quantity negative nine,
plus or minus 13, divided by four.
So one solution is X equals to quantity
negative nine, plus 13, divided by four.
Well negative nine plus 13 equals positive four.
This simplifies to four divided by four, which equals one.
Or, we have X equals the quantity negative nine
minus 13, divided by four,
which equals negative 22, divided by four,
which equals negative 5.5.
So both of these are solutions
to the algebraic equation shown here, or here.
However, we know X is the length,
and therefore the length cannot be negative,
which means for the application problem,
our solution is X equals one.
Going back to our first slide we now know
if we have 22 square feet of fabric to form a border
around a three foot by six foot quilt,
the width of the border should be one foot.
I hope you found this helpful.