x^2 + 1/x^2 = 47. find the value of x+ 1/x and x- 1/x
(x + 1/x)^2 = x^2 + 2 + 1/x^2
(x - 1/x)^2 = x^2 - 2 + 1/x^2
So, letting
u = x + 1/x
v = 1 - 1/x
u^2 + v^2 = 2(x^2 + 1/x^2) = 98
u^2 - v^2 = 4
u^2 = 49
v^2 = 45
x + 1/x = ±7
x - 1/x = ±3√5
oops. 94, not 98
To find the values of x + 1/x and x - 1/x, let's start by manipulating the given equation:
x^2 + 1/x^2 = 47
To simplify the equation, we can multiply both sides by x^2, giving us:
x^4 + 1 = 47x^2
Now, let's rearrange the equation to form a quadratic equation:
x^4 - 47x^2 + 1 = 0
Let's replace x^2 with a variable, say, y:
y^2 - 47y + 1 = 0
Now, using the quadratic formula, we can solve for y:
y = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -47, and c = 1. Substituting these values into the formula, we have:
y = (47 ± √((-47)^2 - 4(1)(1))) / 2(1)
Simplifying further:
y = (47 ± √(2209 - 4)) / 2
y = (47 ± √2205) / 2
Since we're looking for real values, we take the positive square root, so:
y = (47 + √2205) / 2
Note: The negative square root gives us imaginary solutions.
Now, we can use the value of y to find x in terms of y:
x^2 = y
x = ±√y
Therefore:
x = ±√[(47 + √2205) / 2]
Now, let's calculate the values of x + 1/x and x - 1/x:
x + 1/x = ±√[(47 + √2205) / 2] + 1 / ±√[(47 + √2205) / 2]
x - 1/x = ±√[(47 + √2205) / 2] - 1 / ±√[(47 + √2205) / 2]
Since we have a choice of two signs for x (±), we will have a total of four possible combinations to find the values of x + 1/x and x - 1/x.