2-(x-5)2%3D9.jpeg "(x+4)2-(x-5)2=9")
Solving the Equation: (x+4)² - (x-5)² = 9
This article will guide you through the process of solving the equation (x+4)² - (x-5)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:
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Recognize the pattern: Notice that both (x+4)² and (x-5)² are perfect squares.
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Apply the formula: Using the difference of squares factorization, we can rewrite the equation as:
(x+4 + x-5)(x+4 - (x-5)) = 9
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Simplify: Combining like terms, we get:
(2x - 1)(9) = 9
Solving for x
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Divide both sides by 9:
2x - 1 = 1
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Add 1 to both sides:
2x = 2
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Divide both sides by 2:
x = 1
Solution
Therefore, the solution to the equation (x+4)² - (x-5)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.