The expression you've given is:

\[
a^4 - 27a^2 + 1
\]

This is a quartic polynomial (a polynomial of degree 4). Since you haven't provided a specific question, I'll assume you're looking for how to factor it or find its roots.

Let's attempt to factor the expression \( a^4 - 27a^2 + 1 \).

1. **Substitution method**: Set \( x = a^2 \), which means that \( a^4 = x^2 \). So, the expression becomes a quadratic in \( x \):

\[
x^2 - 27x + 1
\]

2. **Solve using the quadratic formula**: The quadratic formula is given by:

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

Here, \( a = 1 \), \( b = -27 \), and \( c = 1 \).

\[
x = \frac{-(-27) \pm \sqrt{(-27)^2 - 4(1)(1)}}{2(1)}
\]
\[
x = \frac{27 \pm \sqrt{729 - 4}}{2}
\]
\[
x = \frac{27 \pm \sqrt{725}}{2}
\]

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

\[
x_1 = \frac{27 + \sqrt{725}}{2}, \quad x_2 = \frac{27 - \sqrt{725}}{2}
\]

3. **Back-substitute**: Recall that \( x = a^2 \), so to find \( a \), take the square root of both \( x_1 \) and \( x_2 \):

\[
a_1 = \pm \sqrt{x_1}, \quad a_2 = \pm \sqrt{x_2}
\]

This provides the roots of the equation in terms of \( a \).

Would you like further details on factoring this expression or working through these roots?

### Related questions:
1. How do you simplify expressions involving square roots?
2. What is the quadratic formula and how is it derived?
3. How can you verify if a quartic equation has real or complex roots?
4. What are some methods for solving quartic polynomials directly?
5. How do substitutions help in solving higher degree polynomials?

**Tip**: Always check if an equation can be simplified by factoring or using substitutions before resorting to the quadratic formula.

Explore how to factor and solve the quartic polynomial a⁴ - 27a² + 1 using the substitution method and the quadratic formula. This guide provides a step-by-step approach to identifying the roots of the equation, making it suitable for high school students studying advanced algebra concepts.

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