Solve the following equation using the quadratic formula. [4 + (1/x) - (1/x^2) = 0]

We are given the equation:

\[
4 + \frac{1}{x} - \frac{1}{x^2} = 0
\]

To solve this equation using the quadratic formula, we need to first rewrite it in a standard quadratic form. Let's make a substitution:

Let \( y = \frac{1}{x} \), so \( \frac{1}{x^2} = y^2 \).

Substituting into the equation, we get:

\[
4 + y - y^2 = 0
\]

Rearrange it to fit the standard quadratic form:

\[
-y^2 + y + 4 = 0
\]

Multiplying through by \(-1\) to simplify:

\[
y^2 - y - 4 = 0
\]

Now, we can apply the quadratic formula, which is:

\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( y^2 - y - 4 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -4 \)

Substitute these values into the quadratic formula:

\[
y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-4)}}{2(1)}
\]

Simplifying:

\[
y = \frac{1 \pm \sqrt{1 + 16}}{2}
\]
\[
y = \frac{1 \pm \sqrt{17}}{2}
\]

So the two solutions for \( y \) are:

\[
y = \frac{1 + \sqrt{17}}{2} \quad \text{or} \quad y = \frac{1 - \sqrt{17}}{2}
\]

Since \( y = \frac{1}{x} \), we can now find \( x \):

\[
x = \frac{1}{y}
\]

Thus, the two values for \( x \) are:

\[
x = \frac{2}{1 + \sqrt{17}} \quad \text{or} \quad x = \frac{2}{1 - \sqrt{17}}
\]

We can rationalize these solutions if needed, but these are the exact forms of \( x \).

Would you like further details on rationalizing the solutions or have any questions?

Here are some related questions to explore:
1. How do you rationalize expressions like \( \frac{2}{1 + \sqrt{17}} \)?
2. What is the significance of the discriminant \( \sqrt{b^2 - 4ac} \) in the quadratic formula?
3. How does the quadratic formula change if \( a \neq 1 \)?
4. Can we solve this equation by factoring instead of using the quadratic formula?
5. How does the substitution method work in transforming complex equations?

**Tip:** When using the quadratic formula, always check if simplifying the equation or substituting can make calculations easier!

Learn how to solve the equation 4 + (1/x) - (1/x²) = 0 using the quadratic formula and substitution method. This step-by-step solution demonstrates how to transform the equation into standard quadratic form, apply the quadratic formula, and find exact solutions. Ideal for students in Grades 9-12.

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