A rectangular garden has length twice as great as its width. A second rectangular garden has the same width as the first garden and length that is 4 meters greater than the length of the first garden. The second garden has area of 70 square meters. What is the width of the two gardens? Enter your answer in the box.

The width is 5 meters.

Let w = the width of the gardens.  In the first garden, the length is twice the width, so l = 2w.  

In the second garden, the length is 4 more than the first garden, so l = 2w+4.

The area of a rectangle is found by multiplying the length and width; for the second garden, that is 

w(2w+4) = 70

Using the distributive property, 
w*2w+w*4 = 70
2w² + 4w = 70

We can factor a 2 out of the left hand side:
2(w² + 2w) = 70

Divide both sides by 2, and we have
w² + 2w = 35

We want the equation equal to 0 to find the roots (solutions), so subtract 35 from both sides:
w² + 2w - 35 = 35 - 35
w² + 2w - 35 = 0

This is easily factorable; we want factors of -35 that sum to 2.  7(-5) = -35 and 7 + (-5) = 2, so
(w+7)(w-5) = 0

Using the zero product property, we know that either w+7=0 or w-5=0; this gives us w=-7 or w=5. Since a negative width makes no sense, we know that w=5.