To solve the equation \(\frac{-9 + (x - 3)^2}{-4} = -4\), we first eliminate the fraction by multiplying both sides by \(-4\):
\[ -9 + (x - 3)^2 = 16 \]
Next, we isolate \((x - 3)^2\) by adding \(9\) to both sides of the equation:
\[ (x - 3)^2 = 16 + 9 \] \[ (x - 3)^2 = 25 \]
Now, we can solve for \(x - 3\) by taking the square root of both sides. Remember to consider both the positive and negative roots:
\[ x - 3 = \pm 5 \]
This gives us two cases to consider:
- \(x - 3 = 5\)
- \(x - 3 = -5\)
Now we solve each case:
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For \(x - 3 = 5\): \[ x = 5 + 3 = 8 \]
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For \(x - 3 = -5\): \[ x = -5 + 3 = -2 \]
Thus, the solutions to the quadratic equation are:
\[ x = 8 \quad \text{and} \quad x = -2 \]
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