Let's start with what we know:
The area of the rectangle is 54cm² which means: length x width = 54cm²
For the sake of this problem let's refer to length as l and width as w.
We are also trying to find x so let's set up some equations based on the info above.
You said that the length is 2cm more than x. That can be written as this:
l = 2 + x
And you said that the width is 5cm less than twice x. That be written as this:
w = 2x - 5
Let's take those two equations and substitute them into our very first equation (l x w = 54cm²)
l x w = 54cm²
l = 2 + x and w = 2x - 5, so:
(2 + x) x (2x - 5) = 54
Let's use the FOIL method to flesh this out. Remember that FOIL stands for first, outer, inner, last:
First: (2 × 2x)
Outer: (2 × -5)
Inner: (x × 2x)
Last: (x × -5)
That would appear in the equation like this:
(2 × 2x) + (2 × -5) + (x × 2x) + (x × -5) = 54
Time to simplify!
4x - 10 + 2x² - 5x = 54
Which can be simplified further to:
2x² - x - 10 = 54
Subtract 54 from each side:
2x² - x - 10 - 54 = 54 -54
And simplify:
2x² - x - 64 = 0
That's a quadratic equation which means we can solve for x using the quadratic formula which is this:
-b
Looking at <em>our </em>equation we know that a = 2, b = -1, and c = -64 so let's plug those values in!
Time to simplify:
That's a messy number, but it's what we got! So let's do the + / -
(1 + 22.65)/4 ≈ 5.91
(1 - 22.65)/4 ≈ -5.41
So after the quadratic formula, we believe that x ≈either 5.91 or -5.41
Let's test the two to see which one works.
Remember our earlier equations? Let's use those here.
l = 2 + x, so:
l ≈ 2 + 5.91
l ≈ 7.91cm
And w = 2x - 5, so:
w ≈ 2(5.91) - 5
w ≈ 11.82 - 5
w ≈ 6.82cm
Let's see if that come's close to our area of 54cm²
l x w = 54cm²
7.91cm x 6.82cm = 53.95cm²
If we round that up, it will equal 54cm² so we know that the x-value of 5.91 is the correct answer without even having to test -5.41.
Answer: x ≈
5.91
