To solve the equation \( x + 8 = \sqrt{12x - 11} + 6 \), we start by isolating the square root term.
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Rearranging the equation yields: \[ \sqrt{12x - 11} = x + 8 - 6 \] \[ \sqrt{12x - 11} = x + 2 \]
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Next, we square both sides to eliminate the square root: \[ (\sqrt{12x - 11})^2 = (x + 2)^2 \] \[ 12x - 11 = x^2 + 4x + 4 \]
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Rearranging gives: \[ 0 = x^2 + 4x + 4 - 12x + 11 \] \[ 0 = x^2 - 8x + 15 \]
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This is a quadratic equation. We can factor it: \[ 0 = (x - 3)(x - 5) \]
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Setting each factor equal to 0 gives the potential solutions: \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
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We need to check both solutions in the original equation to see if they are valid.
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For \( x = 3 \): \[ 3 + 8 = \sqrt{12(3) - 11} + 6 \] \[ 11 = \sqrt{36 - 11} + 6 \] \[ 11 = \sqrt{25} + 6 \] \[ 11 = 5 + 6 \quad \text{(true)} \]
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For \( x = 5 \): \[ 5 + 8 = \sqrt{12(5) - 11} + 6 \] \[ 13 = \sqrt{60 - 11} + 6 \] \[ 13 = \sqrt{49} + 6 \] \[ 13 = 7 + 6 \quad \text{(true)} \]
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Both solutions are valid. Therefore, the solution to the equation is: \[ x = 3 \text{ and } x = 5 \]
So the correct response is: \( x = 3 \text{ and } x = 5 \).