SOLUTION: If w is equal to 1cm in this : w:l equals l:w+l What should be the equivalent of "l"?

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Question 958167: If w is equal to 1cm in this : w:l equals l:w+l
What should be the equivalent of "l"?
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
.
For clarity, let x=l (to prevent confusion with one)
w%2Fx=x%2F%28w%2Bx%29 Put value in for w:
1cm%2Fx=x%2F%281cm%2Bx%29 Multiply by x
1cm=%28x%5E2%29%2F%281cm%2Bx%29 Multiply by 1cm+x
x%2B1=x%5E2 Subtract (x+1)from each side.
0=x%5E2-x-1
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-1x%2B-1+=+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=%28-1%29%5E2-4%2A1%2A-1=5.

Discriminant d=5 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1%2B-sqrt%28+5+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1%29%2Bsqrt%28+5+%29%29%2F2%5C1+=+1.61803398874989
x%5B2%5D+=+%28-%28-1%29-sqrt%28+5+%29%29%2F2%5C1+=+-0.618033988749895

Quadratic expression 1x%5E2%2B-1x%2B-1 can be factored:
1x%5E2%2B-1x%2B-1+=+1%28x-1.61803398874989%29%2A%28x--0.618033988749895%29
Again, the answer is: 1.61803398874989, -0.618033988749895. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1%2Ax%2B-1+%29

ANSWER: x=1.62 or -0.62 (Remember x is l)