To solve the equation \( 6n^2 - 5n - 12 = 18 \) by completing the square, we will first rearrange it into standard form and then complete the square.
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Move \( 18 \) to the left side of the equation:
\[ 6n^2 - 5n - 12 - 18 = 0 \]
This simplifies to:
\[ 6n^2 - 5n - 30 = 0 \]
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Divide every term by \( 6 \) to simplify:
\[ n^2 - \frac{5}{6}n - 5 = 0 \]
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Add \( 5 \) to both sides:
\[ n^2 - \frac{5}{6}n = 5 \]
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To complete the square, take half of the coefficient of \( n \) (which is \( -\frac{5}{6} \)), square it, and add it to both sides. Half of \( -\frac{5}{6} \) is \( -\frac{5}{12} \), and squaring gives:
\[ \left(-\frac{5}{12}\right)^2 = \frac{25}{144} \]
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Add \( \frac{25}{144} \) to both sides:
\[ n^2 - \frac{5}{6}n + \frac{25}{144} = 5 + \frac{25}{144} \]
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The left side can now be expressed as a square:
\[ \left(n - \frac{5}{12}\right)^2 = 5 + \frac{25}{144} \]
To combine the right side, convert \( 5 \) into a fraction over \( 144 \):
\[ 5 = \frac{720}{144} \] So,
\[ 5 + \frac{25}{144} = \frac{720 + 25}{144} = \frac{745}{144} \]
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We now have:
\[ \left(n - \frac{5}{12}\right)^2 = \frac{745}{144} \]
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Take the square root of both sides:
\[ n - \frac{5}{12} = \pm \sqrt{\frac{745}{144}} \]
Thus:
\[ n - \frac{5}{12} = \pm \frac{\sqrt{745}}{12} \]
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Solve for \( n \):
\[ n = \frac{5}{12} \pm \frac{\sqrt{745}}{12} \]
This can be expressed as:
\[ n = \frac{5 \pm \sqrt{745}}{12} \]
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Now we can compute the numerical values. First, find \( \sqrt{745} \):
\[ \sqrt{745} \approx 27.36 \]
So,
\[ n_1 = \frac{5 + 27.36}{12} \approx \frac{32.36}{12} \approx 2.6967 \approx 2.70 \quad (\text{to the nearest hundredth}) \] \[ n_2 = \frac{5 - 27.36}{12} \approx \frac{-22.36}{12} \approx -1.8633 \approx -1.86 \quad (\text{to the nearest hundredth}) \]
The final solutions for \( n \) are:
\[ \boxed{2.70, -1.86} \]
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