Bot GPT 4-o mini To determine how many solutions are in the solution set for the equation \( 2(x - 5)^2 + 2 = 20 \), we can first simplify and solve the equation step by step.
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Start by isolating the term with the square: \[ 2(x - 5)^2 + 2 = 20 \] Subtract 2 from both sides: \[ 2(x - 5)^2 = 18 \] Now divide both sides by 2: \[ (x - 5)^2 = 9 \]
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Next, take the square root of both sides: \[ x - 5 = \pm 3 \]
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This gives us two separate equations to solve: \[ x - 5 = 3 \quad \text{and} \quad x - 5 = -3 \]
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Solving these equations:
- For \( x - 5 = 3 \): \[ x = 8 \]
- For \( x - 5 = -3 \): \[ x = 2 \]
Thus, the solutions are \( x = 2 \) and \( x = 8 \).
Therefore, there are two solutions in the solution set for the original equation.
The correct response is: two
1 answer