To find the number of solutions for the equation \( 2 (x-5)^2 + 2 = 20 \), we will first simplify the equation step by step.
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Start with the original equation: \[ 2 (x-5)^2 + 2 = 20 \]
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Subtract 2 from both sides: \[ 2 (x-5)^2 = 18 \]
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Divide both sides by 2: \[ (x-5)^2 = 9 \]
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Now, take the square root of both sides: \[ x - 5 = 3 \quad \text{or} \quad x - 5 = -3 \]
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Solve for \( x \):
- For the equation \( x - 5 = 3 \): \[ x = 3 + 5 = 8 \]
- For the equation \( x - 5 = -3 \): \[ x = -3 + 5 = 2 \]
Thus, we have two solutions: \( x = 8 \) and \( x = 2 \).
Conclusion: There are 2 solutions in the solution set.