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The image contains two mathematical expressions:

1. \( x^2 - 5x + 1 = 0 \) (a quadratic equation)
2. \(\frac{x^{16} - 1}{x^8}\)

### Let's address these one by one:

### 1. Solving the quadratic equation:
We are given the equation \(x^2 - 5x + 1 = 0\). To solve it, we can use the quadratic formula:

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

Here, \(a = 1\), \(b = -5\), and \(c = 1\). Substituting into the formula:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(1)}}{2(1)}
\]
\[
x = \frac{5 \pm \sqrt{25 - 4}}{2}
\]
\[
x = \frac{5 \pm \sqrt{21}}{2}
\]

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

\[
x_1 = \frac{5 + \sqrt{21}}{2}, \quad x_2 = \frac{5 - \sqrt{21}}{2}
\]

### 2. Simplifying the expression \(\frac{x^{16} - 1}{x^8}\):
We are given the expression \(\frac{x^{16} - 1}{x^8}\).

This can be simplified by factoring the numerator:

\[
\frac{x^{16} - 1}{x^8} = \frac{(x^8 - 1)(x^8 + 1)}{x^8}
\]

Now, we can cancel the \(x^8\) term from the numerator and denominator:

\[
= x^8 - 1
\]

Thus, the simplified form of the expression is \(x^8 - 1\).

---

Would you like more details or explanations for any part of the solution?

### Related questions:
1. Can the quadratic equation \(x^2 - 5x + 1 = 0\) be solved using a different method?
2. How would you factor other polynomial expressions like \(x^{16} - 1\)?
3. What does the quadratic formula help us understand about the nature of the roots?
4. What are the applications of simplifying expressions like \(\frac{x^{16} - 1}{x^8}\) in real-world problems?
5. How would the solution change if the quadratic equation had complex roots?

### Tip:
For quadratic equations, the discriminant (\(b^2 - 4ac\)) tells you whether the roots are real or complex. If the discriminant is positive, the roots are real and distinct. If it is zero, the roots are real and equal. If negative, the roots are complex.

Solve the quadratic equation x^2 - 5x + 1 = 0 and simplify the expression (x^16 - 1) / x^8.

This page explains how to solve the quadratic equation x^2 - 5x + 1 = 0 using the quadratic formula and how to simplify the expression (x^16 - 1) / x^8 through polynomial factorization. It provides a detailed solution suitable for students in Grades 9-12.

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