SOLUTION: Please help me solve plese show step by step I think the answer is x= 19ft 9 in John increased the area of his garden by 120 ft2. The original garden was 12 ft. by 16 ft., and h

Algebra ->  Matrices-and-determiminant -> SOLUTION: Please help me solve plese show step by step I think the answer is x= 19ft 9 in John increased the area of his garden by 120 ft2. The original garden was 12 ft. by 16 ft., and h      Log On


   

Question 467052: Please help me solve plese show step by step I think the answer is x= 19ft 9 in
John increased the area of his garden by 120 ft2. The original garden was 12 ft. by 16 ft., and he increased the length and the width by the same amount. Find the exact dimensions of the new garden and approximate the dimensions in feet and inches. Discuss which method you used to solve the problem and why you chose this method.

Found 2 solutions by Alan3354, Theo:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
John increased the area of his garden by 120 ft2. The original garden was 12 ft. by 16 ft., and he increased the length and the width by the same amount. Find the exact dimensions of the new garden and approximate the dimensions in feet and inches.
The original area was 12*16 = 192 sq ft
192 + 120 = 312 sq ft
(12+x)*(16+x) = 312
x%5E2+%2B+28x+%2B+192+=+312
x%5E2+%2B+28x+-+120+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B28x%2B-120+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2828%29%5E2-4%2A1%2A-120=1264.

Discriminant d=1264 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-28%2B-sqrt%28+1264+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2828%29%2Bsqrt%28+1264+%29%29%2F2%5C1+=+3.77638883463118
x%5B2%5D+=+%28-%2828%29-sqrt%28+1264+%29%29%2F2%5C1+=+-31.7763888346312

Quadratic expression 1x%5E2%2B28x%2B-120 can be factored:
1x%5E2%2B28x%2B-120+=+%28x-3.77638883463118%29%2A%28x--31.7763888346312%29
Again, the answer is: 3.77638883463118, -31.7763888346312. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B28%2Ax%2B-120+%29

Ignore the neg solution
x = sqrt(316) - 14 feet added to 12 & 16
x = 3.776 ft
--------------
You can convert to inches, etc

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your original garden is 12 * 16 = 192 square feet.
the width is 12 and the length is 16.
you added 120 square feet to it.
the new garden is 192 + 120 = 312 square feet.
you added the same amount to the length and the width.
we'll call that x.
your formula for the new garden is:
(x + 12) * (x + 16) = 312
multiply out the factors to get:
x^2 + 28x + 192 = 312
we'll solve by completing the squares.
subtract 192 from both sides of this equation to get:
x^2 + 28x = 120
take 1/2 of 28 and square it and then add it to the right side of the equation.
you'll get:
x^2 + 28x = 120 + 196
this becomes:
x^2 + 28x = 316
take 1/2 of 28 and form the squaring factor of:
(x + 14)^2 = 316
that's your completing the squares equation.
take the square root of both sides of this equation to get:
x + 14 = +/- sqrt(316)
subtract 14 from both sides of this equation to get:
x = +/- sqrt(316) - 14
solve for x to get:
x = 3.776388835
or:
x = -31.77638883
since x can't be negative, you are left with:
x = 3.776388835
replace x in your original equation for the new garden with that number and you get:
(x + 12) * (x + 16) = 312 becomes:
(3.776388835 + 12) * (3.776388835 + 16) = 312
simplify that equation to get:
(15.776388835) * (19.776388835) = 312
multiply that out and you'll get:
312 = 312, confirming that the value for x is good.
the exact dimensions of the new garden are:
length = 19.776388835
width = 15.776388835
to convert this to feet and inches, multiply the fractional part by 12 and add that number of inches to the integral part, which is the feet.
you will get:
length = 19 feet 9.3166660915 inches
width = 15 feet 9.3166660915 inches
you can round to whatever decimal place on the inches that you want.
looking at your answer, it looks like you got one dimension ok, but you forgot to get the other dimension.
you got what i call the length.
you need to add the same amount you added to the length, to the width, to get 15 feet 9 inches.
you should have added 3.776388835 feet which became close to 3 feet 9 inches as you showed, if you are rounding to the nearest inch.