9514 1404 393
Answer:
a) x = (√5 -1)/2 ≈ 0.618034
b) 1/x = (√5 +1)/2 ≈ 1.618034
Step-by-step explanation:
Given:
x/(1 -x) = 1/x
Find:
Exactly, and as a decimal approximation, ...
a) x, using the quadratic formula
b) 1/x
Solution:
a) We can multiply the given equation by x(1 -x) to obtain ...
x² = 1 -x
x² +x -1 = 0 . . . . . . add x-1
The coefficients for use in the quadratic formula are a=1, b=1, c=-1. The solution using the quadratic formula is ...
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{-1\pm\sqrt{1^2-4\cdot1\cdot(-1)}}{2\cdot1}=\dfrac{-1\pm\sqrt{5}}{2}[/tex]
We are only interested in the positive solution, which is ...
[tex]\boxed{x=\dfrac{\sqrt{5}-1}{2}\approx0.618034}[/tex]
__
b) The quadratic we developed in the first part can be rearranged like this:
x² +x = 1 . . . . . . add 1 to both sides
x(x +1) = 1 . . . . . factor out x
x +1 = 1/x . . . . . .divide by x
Then to find the value of 1/x, we simply need to add 1 to the value of x we have:
[tex]\dfrac{1}{x}=\dfrac{\sqrt{5}-1}{2}+1=\dfrac{\sqrt{5}-1}{2}+\dfrac{2}{2}=\dfrac{\sqrt{5}-1+2}{2}\\\\\boxed{\dfrac{1}{x}=\dfrac{\sqrt{5}+1}{2}\approx1.618034}[/tex]
