If the value of x+1/x=2 then what is the value of x^2014+1/x^2016
To find the value of x^2014 + 1/x^2016, we should first simplify the expression x + 1/x = 2.
Multiplying both sides of the equation by x, we get:
x(x + 1/x) = 2x
Expanding the left side of the equation:
x^2 + 1 = 2x
Rearranging the terms to a quadratic equation form:
x^2 - 2x + 1 = 0
Now, we can solve this quadratic equation. There are multiple ways to solve quadratic equations, but one common way is to use the quadratic formula:
For an equation in the form ax^2 + bx + c = 0, the quadratic formula states that the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = -2, and c = 1. Plugging in these values into the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4(1)(1))) / (2(1))
x = (2 ± √(4 - 4)) / 2
x = (2 ± √(0)) / 2
Since the discriminant (b^2 - 4ac) is zero, there is only one solution for x:
x = 2 / 2
x = 1
Therefore, the value of x is 1.
Now we can substitute this value of x into the expression x^2014 + 1/x^2016:
1^2014 + 1/1^2016
1 + 1/1
1 + 1
= 2
So, the value of x^2014 + 1/x^2016, when x + 1/x = 2, is 2.