SOLUTION: Please help with the following word problem. Thank you!! The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 4 cm, what are the dimens

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Please help with the following word problem. Thank you!! The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 4 cm, what are the dimens      Log On


   

Question 42460: Please help with the following word problem. Thank you!!
The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 4 cm, what are the dimensions (the length and the width) of the rectangle?

Answer by Nate(3500) About Me  (Show Source):
You can put this solution on YOUR website!
length = 1 + w
width = w
Pythagoras Theorm:
a%5E2+%2B+b%5E2+=+c%5E2
%281+%2B+w%29%5E2+%2B+%28w%29%5E2+=+%284%29%5E2
1+%2B+2w+%2B+w%5E2+%2B+w%5E2+=+16
2w%5E2+%2B+2w+-+15+=+0
a=2 b=2 c=-15
w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
w+=+%28-2+%2B-+sqrt%28+2%5E2-4%2A2%2A%28-15%29+%29%29%2F%282%2A2%29+
w+=+%28-2+%2B-+sqrt%28+124+%29%29%2F%284%29+
w+=+%28-1%2F2%29+%2B-+sqrt%28124%29%2F4
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 2w%5E2%2B2w%2B-15+=+0) has the following solutons:

w%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=%282%29%5E2-4%2A2%2A-15=124.

Discriminant d=124 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+124+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%282%29%2Bsqrt%28+124+%29%29%2F2%5C2+=+2.28388218141501
w%5B2%5D+=+%28-%282%29-sqrt%28+124+%29%29%2F2%5C2+=+-3.28388218141501

Quadratic expression 2w%5E2%2B2w%2B-15 can be factored:
2w%5E2%2B2w%2B-15+=+2%28w-2.28388218141501%29%2A%28w--3.28388218141501%29
Again, the answer is: 2.28388218141501, -3.28388218141501. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B2%2Ax%2B-15+%29

Answer: w+=+%28-1%2F2%29+%2B+sqrt%28124%29%2F4
length: l+=+1+%2B+%28-1%2F2%29+%2B+sqrt%28124%29%2F4