Solving the equation 1/p_1 = 2p - 4

To solve the given equation 1/p_1 = 2p - 4, we need to isolate the variable p_1 on one side of the equation. Let's go step by step to solve it.

Step 1: Simplify the equation
First, we can simplify the right side of the equation by combining like terms. The equation becomes:

1/p_1 = 2p - 4

Step 2: Clear the fraction
To clear the fraction, we can multiply both sides of the equation by p_1. This will eliminate the fraction on the left side. The equation becomes:

1 = (2p - 4) * p_1

Step 3: Distribute the terms
Next, we need to distribute the terms on the right side of the equation. This can be done by multiplying each term inside the parentheses by p_1:

1 = 2p * p_1 - 4 * p_1

Simplifying further, we have:

1 = 2p^2 - 4p_1

Step 4: Rearrange the equation
Now, let's rearrange the equation to isolate the variable p_1. We can move the terms to one side of the equation:

2p^2 - 4p_1 - 1 = 0

This equation is a quadratic equation in terms of p_1. To solve it, we can use the quadratic formula:

p_1 = (-b ± √(b^2 - 4ac)) / (2a)

Comparing this with our equation, we have:
a = 2
b = -4
c = -1

Step 5: Substitute the values into the quadratic formula
Now, let's substitute the values of a, b, and c into the quadratic formula and simplify:

p_1 = (-(-4) ± √((-4)^2 - 4 * 2 * (-1))) / (2 * 2)
= (4 ± √(16 + 8)) / 4
= (4 ± √24) / 4
= (4 ± √(4 * 6)) / 4
= (4 ± 2√6) / 4

Step 6: Simplify the solution
Finally, let's simplify the solution:

p_1 = (4 + 2√6) / 4
= 1 + 0.5√6

or

p_1 = (4 - 2√6) / 4
= 1 - 0.5√6

Conclusion
The solution to the equation 1/p_1 = 2p - 4 is p_1 = 1 + 0.5√6 or p_1 = 1 - 0.5√6. These are the two possible values for p_1 that satisfy the given equation.