Example 5 Solve a Quadratic Equation by Factoring a Difference of Squares Solve $$ 49-x^{2}=0 $$. Check your solution.

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1. Write the equation:  
   $$
   49 - x^2 = 0
   $$

2. Recognize the difference of squares form:  
   $$
   49 - x^2 = (7)^2 - x^2
   $$  
   which can be factored as  
   $$
   (7 - x)(7 + x).
   $$

3. Set the factored expression equal to zero:  
   $$
   (7 - x)(7 + x) = 0.
   $$

4. Apply the Zero Product Property by setting each factor equal to zero:  
   $$
   7 - x = 0 \quad \text{or} \quad 7 + x = 0.
   $$

5. Solve each equation:

- For $$ 7 - x = 0 $$:  
     $$
     x = 7.
     $$
   
   - For $$ 7 + x = 0 $$:  
     $$
     x = -7.
     $$

6. Check the solutions:

- Substitute $$ x = 7 $$ into the original equation:
     $$
     49 - (7)^2 = 49 - 49 = 0.
     $$
   
   - Substitute $$ x = -7 $$ into the original equation:
     $$
     49 - (-7)^2 = 49 - 49 = 0.
     $$

Both solutions satisfy the equation.

The solutions are $$ x = 7 $$ and $$ x = -7 $$.
The solutions are $$ x = 7 $$ and $$ x = -7 $$.

Example 5 Solve a Quadratic Equation by Factoring a Difference of Squares Solve \( 49-x^{2}=0 \). Check your solution.

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