On a 15-mile race, runners start with a 9-mile road race followed by 12 laps
around a track that is x miles long.
What is the length of the track? Write
and solve an equation.

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Step-by-step explanation:

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 our 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.

 x ≈ 5.91