To prove that \(a^2 + \frac{1}{a^2} = 227\), given that \(a - \frac{1}{a} = 15\), follow these steps:

1. Start with the given equation:
\[ a - \frac{1}{a} = 15 \]

2. Square both sides of the equation to find an expression for \(a^2 + \frac{1}{a^2}\):
\[ \left(a - \frac{1}{a}\right)^2 = 15^2 \]

3. Expand the left side of the equation:
\[ a^2 - 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^2} = 225 \]

4. Simplify the expression:
\[ a^2 - 2 + \frac{1}{a^2} = 225 \]

5. Rearrange the equation to isolate \(a^2 + \frac{1}{a^2}\):
\[ a^2 + \frac{1}{a^2} - 2 = 225 \]

6. Add 2 to both sides to solve for \(a^2 + \frac{1}{a^2}\):
\[ a^2 + \frac{1}{a^2} = 225 + 2 \]
\[ a^2 + \frac{1}{a^2} = 227 \]

Therefore, the equation is proven as required:
\[ a^2 + \frac{1}{a^2} = 227 \]

If you have any questions or need further details, please let me know!

Here are 5 questions you might consider next:
1. How do you solve quadratic equations using the quadratic formula?
2. What is the process for factoring trinomials?
3. How can you solve equations involving radicals?
4. What are the properties of exponents and how do you apply them?
5. How do you solve systems of linear equations using substitution or elimination methods?

**Tip:** When working with algebraic identities, squaring the equation can often simplify complex terms and reveal the necessary relationships.

Learn how to prove that a² + 1/a² = 227 given that a - 1/a = 15. This detailed solution covers squaring a binomial and algebraic manipulations suitable for high school students in Grades 9-11.

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