To solve the quadratic equation \((x - 3)^2 - 9 = 16\), we'll first simplify it:
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Start by moving 16 to the left side of the equation: \[ (x - 3)^2 - 9 - 16 = 0 \] Which simplifies to: \[ (x - 3)^2 - 25 = 0 \]
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Now, we can rewrite the equation: \[ (x - 3)^2 = 25 \]
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Next, take the square root of both sides: \[ x - 3 = 5 \quad \text{or} \quad x - 3 = -5 \]
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Solving these two equations:
- For \(x - 3 = 5\): \[ x = 5 + 3 = 8 \]
- For \(x - 3 = -5\): \[ x = -5 + 3 = -2 \]
Thus, the solutions to the equation are \(8\) and \(-2\).
Now, listing the smaller solution first, we have: \[ {-2, 8} \]