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Solving the Equation (x + 5)² = 75
This equation involves a squared term, making it a quadratic equation. Here's how to solve it:
1. Isolate the Squared Term
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Take the square root of both sides of the equation: √((x + 5)²) = ±√75
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This simplifies to: x + 5 = ±√75
2. Simplify the Radical
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Find the prime factorization of 75: 75 = 3 x 5 x 5 = 3 x 5²
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Therefore, √75 = √(3 x 5²) = 5√3
3. Solve for x
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Now we have two separate equations:
- x + 5 = 5√3
- x + 5 = -5√3
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Solve for x in each equation:
- x = 5√3 - 5
- x = -5√3 - 5
Solutions
Therefore, the solutions to the equation (x + 5)² = 75 are:
- x = 5√3 - 5
- x = -5√3 - 5
These are the exact solutions. If you need an approximate decimal form, you can use a calculator to evaluate the expressions.