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Solving the Equation: (x + 5)² - 64 = 0
This article will guide you through the process of solving the quadratic equation (x + 5)² - 64 = 0. We will break down the steps and use algebraic techniques to find the solutions.
1. Simplifying the Equation
First, we can simplify the equation by applying the difference of squares factorization:
- (a² - b²) = (a + b)(a - b)
In this case, we have:
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(x + 5)² - 64 = 0
-
(x + 5)² - 8² = 0
Applying the difference of squares factorization:
-
(x + 5 + 8)(x + 5 - 8) = 0
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(x + 13)(x - 3) = 0
2. Finding the Solutions
Now we have a product of two factors that equals zero. This means that at least one of the factors must be equal to zero.
- x + 13 = 0
- x - 3 = 0
Solving for x in each equation:
- x = -13
- x = 3
Conclusion
Therefore, the solutions to the equation (x + 5)² - 64 = 0 are x = -13 and x = 3.
We can verify these solutions by substituting them back into the original equation:
- (-13 + 5)² - 64 = (-8)² - 64 = 64 - 64 = 0
- (3 + 5)² - 64 = (8)² - 64 = 64 - 64 = 0
This confirms that both x = -13 and x = 3 are valid solutions to the given equation.