Answer:
x + 1/x = ± √((3 ± √5) / 2) + 1 / (√((3 ± √5) / 2))
Explanation:
To find the value of x + 1/x, we first need to solve the equation x^2 + 1/(x^2) = 3.
Let's multiply the equation by x^2 to eliminate the fractions:
x^2(x^2) + 1 = 3x^2
This simplifies to:
x^4 + 1 = 3x^2
Rearranging the equation:
x^4 - 3x^2 + 1 = 0
Now, let's substitute y = x^2:
y^2 - 3y + 1 = 0
This is now a quadratic equation in terms of y. We can solve it using the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = -3, and c = 1:
y = (-(-3) ± √((-3)^2 - 4(1)(1))) / (2(1))
Simplifying further:
y = (3 ± √(9 - 4)) / 2
y = (3 ± √5) / 2
Now, let's substitute y back in terms of x:
x^2 = (3 ± √5) / 2
Taking the square root of both sides:
x = ± √((3 ± √5) / 2)
Therefore, the value of x + 1/x can have two possible solutions:
x + 1/x = ± √((3 ± √5) / 2) + 1 / (√((3 ± √5) / 2))