To solve the equation $81x^2 - 16 = 0$ by factoring, follow these steps:

Step 1: Recognize the difference of squares

The given equation is a difference of squares, which follows the general form: $a^2 - b^2 = (a - b)(a + b)$ Here, $81x^2$ is a perfect square, and $16$ is also a perfect square.

We can rewrite the equation as: $(9x)^2 - 4^2 = 0$

Step 2: Factor the equation

Using the difference of squares formula: $(9x - 4)(9x + 4) = 0$

Step 3: Solve for $x$

Set each factor equal to zero and solve for $x$:

1. $9x - 4 = 0$ $9x = 4 \quad \Rightarrow \quad x = \frac{4}{9}$

2. $9x + 4 = 0$ $9x = -4 \quad \Rightarrow \quad x = \frac{-4}{9}$

Final answer:

The solutions are $x = \frac{-4}{9}$ and $x = \frac{4}{9}$. Listed in order from least to greatest:

$x = \frac{-4}{9}, \quad x = \frac{4}{9}$

Would you like further clarification or a detailed explanation of any part of this process?

Here are 5 related questions:

1. How do you solve other quadratic equations by factoring?
2. What is the difference between factoring and using the quadratic formula?
3. Can you factor all types of quadratic equations?
4. How do you identify perfect squares in an equation?
5. What other methods are used to solve quadratic equations?

Tip: Whenever you see a quadratic equation, check for common factoring patterns like the difference of squares or perfect square trinomials!