To solve the equation \( \sqrt{44 - 5x} = x - 6 \), we will follow these steps:
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Square both sides to eliminate the square root: \[ 44 - 5x = (x - 6)^2 \]
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Expand the right side: \[ (x - 6)^2 = x^2 - 12x + 36 \] So the equation becomes: \[ 44 - 5x = x^2 - 12x + 36 \]
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Rearrange the equation: \[ 0 = x^2 - 12x + 36 + 5x - 44 \] This simplifies to: \[ 0 = x^2 - 7x - 8 \]
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Factor the quadratic: We can factor the equation as follows: \[ 0 = (x - 8)(x + 1) \]
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Set each factor to zero:
- \( x - 8 = 0 \) ⇒ \( x = 8 \)
- \( x + 1 = 0 \) ⇒ \( x = -1 \)
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Check for extraneous solutions:
- For \( x = 8 \): \[ \sqrt{44 - 5(8)} = \sqrt{44 - 40} = \sqrt{4} = 2 \] And \( x - 6 = 8 - 6 = 2 \). This solution is valid.
- For \( x = -1 \): \[ \sqrt{44 - 5(-1)} = \sqrt{44 + 5} = \sqrt{49} = 7 \] And \( x - 6 = -1 - 6 = -7 \). This solution is not valid since \( \sqrt{49} \neq -7 \).
Therefore, the only valid solution is: \[ \boxed{8} \]
So the correct response is \( x = 8 \).