Solving Equations: p2 3p - 1 and p2 - 1/p2

In this article, we will delve into the process of solving a quadratic equation and finding the value of a specific expression based on it. The equation we will be working with is ( p^2 3p - 1 ). We will explore various steps and methods to find out the value of ( frac{p^2 - 1}{p^2} ) and related expressions.

Problem Overview

Given the equation ( p^2 3p - 1 ), we need to find the value of ( frac{p^2 - 1}{p^2} ). Let's break this down into manageable steps and find the solution.

Step-by-Step Solution

Step 1: Solve the Quadratic Equation

First, we rewrite the given equation in the standard quadratic form:

[ p^2 - 3p 1 0 ]

To solve for ( p ), we use the quadratic formula: ( p frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a 1 ), ( b -3 ), and ( c 1 ).

Substituting these values, we get:

[ p frac{3 pm sqrt{(-3)^2 - 4(1)(1)}}{2(1)} frac{3 pm sqrt{9 - 4}}{2} frac{3 pm sqrt{5}}{2} ]

Hence, the solutions for ( p ) are:

[ p frac{3 sqrt{5}}{2} quad text{or} quad p frac{3 - sqrt{5}}{2} ]

Step 2: Evaluate ( p^2 - frac{1}{p^2} )

Now let's find the value of ( frac{p^2 - 1}{p^2} ) and ( p^2 - frac{1}{p^2} ).

Starting with ( p^2 - frac{1}{p^2} ), we use the identity:

[ frac{p^2 - 1}{p^2} 1 - frac{1}{p^2} ]

To find ( frac{1}{p^2} ), we note that:

[ p^2 3p - 1 implies p^2 - 1 3p implies 1 - frac{1}{p^2} frac{3p}{p^2} frac{3}{p} ]

So, ( frac{p^2 - 1}{p^2} 1 - frac{3}{p} ).

Next, we calculate ( frac{1}{p^2} ) directly:

[ frac{1}{p^2} frac{1}{(3p - 1)} ]

Combining the above, we get:

[ p^2 - frac{1}{p^2} p^2 - left( frac{1}{p^2} right) frac{p^4 - 1}{p^2} ]

Step 3: Substitute and Simplify

Given ( p^2 3p - 1 ), we substitute this into the expression for ( p^4 - 1 ):

[ p^4 (p^2)^2 (3p - 1)^2 9p^2 - 6p 1 ]

Thus, ( p^4 - 1 9p^2 - 6p ). Hence,

[ frac{p^4 - 1}{p^2} frac{9p^2 - 6p}{p^2} 9 - frac{6p}{p^2} 9 - frac{6}{p} ]

Since ( p frac{3 pm sqrt{5}}{2} ), we have:

[ frac{1}{p} frac{2}{3 pm sqrt{5}} ]

Multiplying the numerator and denominator by the conjugate to rationalize:

[ frac{1}{p} frac{2(3 - sqrt{5})}{(3 sqrt{5})(3 - sqrt{5})} frac{2(3 - sqrt{5})}{9 - 5} frac{2(3 - sqrt{5})}{4} frac{3 - sqrt{5}}{2} ]

Substituting back:

[ 9 - frac{6}{p} 9 - 3(3 - sqrt{5}) 9 - 9 3sqrt{5} 3sqrt{5} ]

Hence, the value of ( frac{p^2 - 1}{p^2} ) is:

[ 1 - frac{3}{p} 1 - 3left( frac{3 - sqrt{5}}{2} right) 1 - frac{9 - 3sqrt{5}}{2} frac{2 - 9 3sqrt{5}}{2} frac{-7 3sqrt{5}}{2} ]

And thus, ( p^2 - frac{1}{p^2} 3sqrt{5} ) and ( frac{p^2 - 1}{p^2} frac{-7 3sqrt{5}}{2} ).

Conclusion

In conclusion, we have demonstrated how to solve the quadratic equation ( p^2 3p - 1 ) and derive the value of ( frac{p^2 - 1}{p^2} ) from it. Through careful algebraic manipulation, we found that ( p^2 - frac{1}{p^2} 3sqrt{5} ).

Understanding these steps provides insight into how to handle quadratic equations and related expressions, a fundamental skill in algebra and higher mathematics.