Solve the given equation by the zero-factor property. $$ x^{2}-64=0 $$

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UpStudy AI · Accepted Answer

To solve the equation $$ x^{2} - 64 = 0 $$ using the **zero-factor property**, follow these steps:

### Step 1: Factor the Equation
The given equation is a **difference of squares**, which can be factored using the formula:
$$ a^{2} - b^{2} = (a - b)(a + b) $$

Apply this to the equation:
$$ x^{2} - 64 = x^{2} - 8^{2} = (x - 8)(x + 8) $$

### Step 2: Apply the Zero-Factor Property
The **zero-factor property** states that if a product of two factors is zero, then at least one of the factors must be zero. Therefore:
$$ (x - 8)(x + 8) = 0 $$

This gives two possible equations:
1. $$ x - 8 = 0 $$
2. $$ x + 8 = 0 $$

### Step 3: Solve for $$ x $$

1. **First Equation:**
   $$ x - 8 = 0 $$
   $$ x = 8 $$

2. **Second Equation:**
   $$ x + 8 = 0 $$
   $$ x = -8 $$

### Conclusion
The solutions to the equation $$ x^{2} - 64 = 0 $$ are:
$$ x = 8 \quad \text{and} \quad x = -8 $$

**Verification:**
Plugging these values back into the original equation:

- For $$ x = 8 $$:
  $$ (8)^2 - 64 = 64 - 64 = 0 $$

- For $$ x = -8 $$:
  $$ (-8)^2 - 64 = 64 - 64 = 0 $$

Both solutions satisfy the equation.

**Final Answer:**
The solutions are $$ x = 8 $$ and $$ x = -8 $$.
The solutions are $$ x = 8 $$ and $$ x = -8 $$.

Solve the given equation by the zero-factor property.

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