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.