Solving for p in q 3 - p^2 / (12p^2)

The equation q 3 - p^2 / (12p^2) may appear complex at first glance, but it can be solved systematically to make p the subject. Let's walk through the process step by step, explore the theory behind it, and discuss the implications of the solution.

Step-by-Step Solution

To solve for p, we need to isolate it on one side of the equation. Here are the detailed steps:

Multiplication to Eliminate the Fraction

First, let's multiply each term by the denominator of the fraction, which is 12p^2 to eliminate the fraction:

q times 12p^2 (3 - p^2) times 12p^2 Expanding the right side: 12qp^2 36p^2 - 12p^4

Rearranging Terms

Next, we rearrange the terms to group like terms together:

12qp^2 12p^4 36p^2 Factor out the common term 12p^2 on the left side: 12p^2(1 p^2) 36p^2

Simplifying and Isolating the Variable

We need to simplify the equation and isolate p^2 on one side:

Divide both sides by 12p^2 (assuming p^2 0): 1 p^2 3 Subtract 1 from both sides: p^2 2

Final Step: Finding p

To find the values of p, we take the square root of both sides:

p pm sqrt{2}

Thus, the final expression for p in terms of q is:

p pm sqrt{frac{3 - q}{12q - 1}}

Decomposition and Interpretation of the Solution

The solution p pm sqrt{frac{3 - q}{12q - 1}} tells us that p depends on the value of q. To ensure the square root is defined, we need:

12q - 1 0, which implies q frac{1}{12} 3 - q 0, which implies q 3

Thus, the domain of q is (frac{1}{12}, 3). If q lies outside this interval, the equation does not provide real solutions for p.

Implications of the Equation

q 3 - p^2 / (12p^2) can be used in various mathematical and scientific contexts. It may represent a relationship between variables in physics, engineering, or economics, for instance. Understanding how to manipulate and solve such equations is crucial in these fields.

Conclusion

By carefully manipulating the given equation, we can express p in terms of q. This process involves algebraic manipulation, recognition of fundamental algebraic identities, and careful consideration of the domain of the solution. Solving such equations is a fundamental skill in mathematics and can be applied to a wide range of practical problems.