To find the number of solutions to the equation \( 2(x - 5)^2 + 2 = 20 \), we can start by simplifying it step by step.
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First, subtract 2 from both sides: \[ 2(x - 5)^2 = 20 - 2 \] \[ 2(x - 5)^2 = 18 \]
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Next, divide both sides by 2: \[ (x - 5)^2 = \frac{18}{2} \] \[ (x - 5)^2 = 9 \]
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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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Solving these equations gives: \[ x - 5 = 3 \implies x = 8 \] \[ x - 5 = -3 \implies x = 2 \]
Thus, the solutions are \( x = 8 \) and \( x = 2 \).
Final Answer: There are two solutions in the solution set.