Solving for x in a Complex Expression to Achieve a Perfect Square

Introduction: Suppose you are dealing with a complex algebraic expression that requires simplification and solving for a variable ( x ) to make it a perfect square. This article will guide you through the process of transforming an expression of the form ( x^2 - frac{1}{x} cdot left(x - frac{1}{x}right) ) into a perfect square.

Expression Simplification

The given expression is:

[left(x^2 - frac{1}{x}right) cdot left(x - frac{1}{x}right)]

First, let's simplify this expression step-by-step.

Step 1: Expand the Expression

[ left(x^2 - frac{1}{x}right) cdot left(x - frac{1}{x}right)]

Step 2: Distribute the Terms

[ x^3 - frac{x^2}{x} - frac{1}{x} cdot x frac{1}{x^2}]

[ x^3 - x - 1 frac{1}{x^2}]

Step 3: Combine Like Terms

[ x^3 - x - 1 frac{1}{x^2}]

We can see that the expression simplifies to:

[x^3 - x - 1 frac{1}{x^2}]

Step 4: Align the Expression for Perfect Square

We want this expression to equal a perfect square. Let's denote a perfect square as ( (x^2 - k)^2 ), where ( k ) is a constant to be determined.

[(x^2 - k)^2 x^4 - 2kx^2 k^2]

Now, we want:

[x^3 - x - 1 frac{1}{x^2} x^4 - 2kx^2 k^2]

This equation will hold if the terms on both sides match. To make the expression a perfect square, we need to find ( k ) and ( x ).

Formulating the Quadratic Equation

To solve for ( x ), we set the simplified expression equal to a perfect square:

[x^2 - x - 1 frac{1}{x^2} m^2]

where ( m ) is a whole number.

Step 1: Rearrange the Equation

[x^2 - x - left(1 - frac{1}{x^2}right) m^2]

This can be further simplified to:

[x^2 - x - 1 frac{1}{x^2} - m^2 0]

Step 2: Convert to a Standard Quadratic Form

[frac{1}{x^2} x^2 - x - 1 - m^2 0]

This is a quadratic equation in terms of ( x ) and can be written as:

[ frac{1}{x^2} (x^2 - x) - (1 m^2) 0]

For simplicity, let's denote the equation as:

[x^2 - x - 1 frac{1}{x^2} - m^2 0]

Step 3: Use the Quadratic Formula

The quadratic formula for a quadratic equation ( ax^2 bx c 0 ) is:

[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Here, our quadratic equation is:

[x^2 - x left(frac{1}{x^2} - m^2 - 1right) 0]

Thus, we have:

[a 1, quad b -1, quad c frac{1}{x^2} - m^2 - 1]

[x frac{1 pm sqrt{1 - 4 cdot 1 cdot left(frac{1}{x^2} - m^2 - 1right)}}{2 cdot 1}]

[x frac{1 pm sqrt{1 - frac{4}{x^2} 4m^2 4}}{2}]

[x frac{1 pm sqrt{5 4m^2 - frac{4}{x^2}}}{2}]

Conclusion

By following these steps, you can solve for ( x ) to make the given expression a perfect square. The key is to align the expression correctly and apply the quadratic formula. Remember, the value of ( x ) will depend on the specific value of ( m ), which can be any whole number.

Related Keywords

Perfect Square, Quadratic Equation, Algebraic Expressions