1. **State the problem:** Explain how we get the equation $x^2 + 1 = 99x$ from the given $x + \frac{1}{x} = 99$. 2. **Start with the given equation:** $$x + \frac{1}{x} = 99$$ 3. **Multiply both sides by $x$:** Since $x \neq 0$, multiply every term by $x$ to eliminate the fraction: $$x \times x + x \times \frac{1}{x} = 99 \times x$$ This simplifies to: $$x^2 + 1 = 99x$$ 4. **Explanation:** - Multiplying $x$ by $x$ gives $x^2$. - Multiplying $x$ by $\frac{1}{x}$ cancels out $x$, leaving $1$. - The right side is simply $99x$. This step is crucial because it transforms the original equation with a fraction into a quadratic form that is easier to work with in algebraic manipulations. **Final result:** $$x^2 + 1 = 99x$$