Solving the Equation: 2p + 1/p = 4 and Finding the Value of p^3 + 1/8p^3
Given Equation: 2p + 1/p = 4
To solve for p, we can start by multiplying both sides of the equation by p, which gives us:
2p^2 + 1 = 4p
Subtracting 4p from both sides and rearranging the terms, we get:
2p^2 - 4p + 1 = 0
This is a quadratic equation, which can be factored as:
(2p - 1)(p - 1) = 0
This tells us that either (2p - 1) = 0 or (p - 1) = 0.
Solving for the first factor, we get:
2p - 1 = 0 --> 2p = 1 --> p = 1/2
And solving for the second factor, we get:
p - 1 = 0 --> p = 1
Therefore, the values of p that satisfy the equation 2p + 1/p = 4 are p = 1/2 and p = 1.
Finding the Value of p^3 + 1/8p^3
Now that we have the values of p, we can find the value of p^3 + 1/8p^3.
For p = 1/2:
p^3 = (1/2)^3 = 1/8
1/8p^3 = 1/8 * 1/8 = 1/64
p^3 + 1/8p^3 = 1/8 + 1/64 = 9/64
For p = 1:
p^3 = 1^3 = 1
1/8p^3 = 1/8 * 1 = 1/8
p^3 + 1/8p^3 = 1 + 1/8 = 9/8
Therefore, the values of p^3 + 1/8p^3 are 9/64 and 9/8, corresponding to the values of p = 1/2 and p = 1, respectively.
