I'm having a hard time figuring out this question,
" There is only one positive number that satisfies the equation
(greek letter phe) - 1 = 1 / (greek letter phe)
Based on this definition, find the value of phe."
Thanks for the help I really appreciate it.
see above
That's phi, not phe
Ø-1 = 1/Ø
Ø^2-Ø-1 = 0
Ø = (1±√5)/2
pick the positive value
To find the value of the Greek letter "phe" (φ) that satisfies the equation φ - 1 = 1/φ, we can follow these steps:
Step 1: Rearrange the equation
Begin by multiplying both sides of the equation by φ to eliminate the fraction:
φ(φ - 1) = 1
Step 2: Expand and simplify
Distribute φ to both terms on the left side of the equation:
φ² - φ = 1
Step 3: Rearrange and set the equation to zero
Move all terms to one side of the equation to obtain a quadratic equation:
φ² - φ - 1 = 0
Step 4: Solve the quadratic equation
At this point, you can either factorize or use the quadratic formula. However, this quadratic equation does not easily factorize, so we will proceed using the quadratic formula.
The quadratic formula states that for an equation in the form ax² + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b² - 4ac)) / 2a
In our case, the equation is φ² - φ - 1 = 0.
By comparing with the quadratic formula, we can identify: a = 1, b = -1, and c = -1.
Substituting the values into the quadratic formula:
φ = (-(-1) ± √((-1)² - 4(1)(-1))) / (2(1))
= (1 ± √(1 + 4)) / 2
= (1 ± √5) / 2
Step 5: Determine the unique positive solution
Since the question asks for the value of φ as a positive number, we need to select only the positive solution. Therefore, the value of φ that satisfies the equation φ - 1 = 1/φ is:
φ = (1 + √5) / 2
So, the value of the Greek letter φ that satisfies the given equation is (1 + √5) / 2.