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Solving the Equation (x+8)(x-8) = -15
This equation presents a classic example of how to solve a quadratic equation by utilizing the difference of squares pattern and applying algebraic manipulations. Let's break down the steps involved:
1. Expand the Left-Hand Side
The left-hand side of the equation is in the form of (a+b)(a-b), which is a common pattern known as the difference of squares. Expanding this pattern gives us:
(x+8)(x-8) = x² - 8² = x² - 64
2. Rewrite the Equation
Now we can rewrite the original equation as:
x² - 64 = -15
3. Isolate the x² Term
To isolate the x² term, add 64 to both sides of the equation:
x² = 49
4. Solve for x
Taking the square root of both sides, we get:
x = ±7
Solution
Therefore, the solutions to the equation (x+8)(x-8) = -15 are x = 7 and x = -7.
Key Takeaways
This example demonstrates the importance of recognizing common algebraic patterns like the difference of squares to simplify equations. By utilizing these patterns and applying basic algebraic manipulations, we can effectively solve quadratic equations and find their solutions.