SOLUTION: Beth's garden is x - 2 feet wide and x +1 feet long. the total area is of the garden 10 square feet. what is the length of the garden in feet.

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Question 1199165: Beth's garden is x - 2 feet wide and x +1 feet long. the total area is of the garden 10 square feet. what is the length of the garden in feet.
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Simple to setup, and may be solved mostly by simple factorization.

Length is 5 feet.

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Assuming x is an integer, then here are the ways to multiply to 10 using only positive integers:
1*10 = 10
2*5 = 10

The gap from x-2 to x+1 is 3 units
Since the gap from -2 to +1 is also 3 units.
Use a number line to see this.

Because of this 3 unit gap, we ignore the factor pair 1*10.
Instead we go for 2*5 which has the 3 unit gap.

x-2 is the smaller of the x expressions.
Meaning the value of x-2 must be 2
This means x+1 must be 5

Answer: The length of the garden is 5 feet long.

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Another approach:

This is probably the more standard universal approach.

The width x-2 and length x+1 are multiplied to get the area of 10 sq ft.

width*length = area
(x-2)*(x+1) = 10
w(x+1) = 10 ...... let w = x-2
wx+w = 10 ...... distribute
x(w) + 1(w) = 10
x(x-2) + 1(x-2) = 10 .... plug in w = x-2
x^2-2x + 1x - 2 = 10
x^2-x-2 = 10
x^2-x-2-10 = 0
x^2-x-12 = 0

We can factor to solve for x
x^2-x-12 = 0
(x-4)(x+3) = 0
x-4 = 0 or x+3 = 0
x = 4 or x = -3

Or we can use the quadratic formula
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-1%29%2B-sqrt%28%28-1%29%5E2-4%281%29%28-12%29%29%29%2F%282%281%29%29

x+=+%281%2B-sqrt%2849%29%29%2F%282%29

x+=+%281%2B-++7%29%2F%282%29

x+=+%281%2B7%29%2F%282%29 or x+=+%281-7%29%2F%282%29

x+=+%288%29%2F%282%29 or x+=+%28-6%29%2F%282%29

x+=+4 or x+=+-3
Ignore the negative x value since it causes either x-2 and x+1 to be negative.
A negative length or width makes no sense.

The only practical solution is x = 4.

If x = 4, then:
x-2 = 4-2 = 2
x+1 = 4+1 = 5
Giving us a 2 ft by 5 ft garden of area 2*5 = 10 sq ft.
The answer is confirmed.

Answer: The length of the garden is 5 feet long.