Solving the Equation: (x+4)²  (x5)² = 9
This article will guide you through the process of solving the equation (x+4)²  (x5)² = 9. We will use the difference of squares factorization to simplify the equation and then solve for x.
Understanding the Difference of Squares
The difference of squares factorization states that a²  b² = (a + b)(a  b). We can apply this to our equation:

Recognize the pattern: Notice that both (x+4)² and (x5)² are perfect squares.

Apply the formula: Using the difference of squares factorization, we can rewrite the equation as:
(x+4 + x5)(x+4  (x5)) = 9

Simplify: Combining like terms, we get:
(2x  1)(9) = 9
Solving for x

Divide both sides by 9:
2x  1 = 1

Add 1 to both sides:
2x = 2

Divide both sides by 2:
x = 1
Solution
Therefore, the solution to the equation (x+4)²  (x5)² = 9 is x = 1.
Verification
We can verify our answer by substituting x = 1 back into the original equation:
(1 + 4)²  (1  5)² = 9 (5)²  (4)² = 9 25  16 = 9 9 = 9
This confirms that x = 1 is indeed the correct solution.