SOLUTION: Help! Don't know how to put story problem into quadratic formula: A rectangular garden is 30 ft by 40 ft. Part of the garden is removed in order to install a walkway of uniform

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Question 833912: Help! Don't know how to put story problem into quadratic formula:
A rectangular garden is 30 ft by 40 ft. Part of the garden is removed in order to install a walkway of uniform width around it. The area of the new garden is one-half the area of the old garden. How wide is the walkway?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
A rectangular garden is 30 ft by 40 ft. Part of the garden is removed in order to install a walkway of uniform width around it. The area of the new garden is one-half the area of the old garden. How wide is the walkway?

Area of the entire garden is A=+30ft+%2A+40ft+=+1200ft%5E2
Let x be the width of the walkway.
Length of the reduced garden is L=+%2840+-+2x%29ft
Width of the reduced garden is +W=+%2830+-+2x%29ft
Area of the reduced garden is A%5Br%5D+=+%2840+-+2x%29+%2A+%2830+-+2x%29ft%5E2

Area of the reduced garden is one-half the area of the old garden; so, we have
A%5Br%5D+=A%2F2 or
2A%5Br%5D+=A
since
A%5Br%5D+=+%2840+-+2x%29+%2A+%2830+-+2x%29ft%5E2 and A=+1200ft%5E2, we have

2%2840+-+2x%29+%2A+%2830+-+2x%29ft%5E2=1200ft%5E2...simplify
%2840+-+2x%29+%2A+%2830+-+2x%29=600....expand
120+-80x-+60x+%2B+4x%5E2=600
+4x%5E2-140x%2B1200=600
+4x%5E2-140x%2B1200-600=0
+4x%5E2-140x-600=0...both sides divide by 4
+x%5E2-35x-150=0......solve for x

Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 4%2Ax%5E2-140%2Ax%2B600=0 ( notice a=4, b=-140, and c=600)





x+=+%28--140+%2B-+sqrt%28+%28-140%29%5E2-4%2A4%2A600+%29%29%2F%282%2A4%29 Plug in a=4, b=-140, and c=600




x+=+%28140+%2B-+sqrt%28+%28-140%29%5E2-4%2A4%2A600+%29%29%2F%282%2A4%29 Negate -140 to get 140




x+=+%28140+%2B-+sqrt%28+19600-4%2A4%2A600+%29%29%2F%282%2A4%29 Square -140 to get 19600 (note: remember when you square -140, you must square the negative as well. This is because %28-140%29%5E2=-140%2A-140=19600.)




x+=+%28140+%2B-+sqrt%28+19600%2B-9600+%29%29%2F%282%2A4%29 Multiply -4%2A600%2A4 to get -9600




x+=+%28140+%2B-+sqrt%28+10000+%29%29%2F%282%2A4%29 Combine like terms in the radicand (everything under the square root)




x+=+%28140+%2B-+100%29%2F%282%2A4%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28140+%2B-+100%29%2F8 Multiply 2 and 4 to get 8


So now the expression breaks down into two parts


x+=+%28140+%2B+100%29%2F8 or x+=+%28140+-+100%29%2F8


Lets look at the first part:


x=%28140+%2B+100%29%2F8


x=240%2F8 Add the terms in the numerator

x=30 Divide


So one answer is

x=30




Now lets look at the second part:


x=%28140+-+100%29%2F8


x=40%2F8 Subtract the terms in the numerator

x=5 Divide


So another answer is

x=5


So our solutions are:

x=30 or x=5



If x+=+5, the length is L=+%2840+-+2%2A5%29+=40-+10=30
and width W=+%2830+-+2%2A5%29+=30-10=20
If x+=+30, the length is L=+%2840+-+2%2A30%29+=40-+60=-20
and width W=+%2830+-+2%2A30%29+=+-30
So, the second case (x+=+30) is ruled out because you can't have a width and a length equal to negative number.

so the walkway is x=5ft wide