To solve the equation
\[ 2 + \frac{x^2}{5} = \frac{x}{6}, \]
first, we can eliminate the fractions by finding a common denominator for the entire equation. The common denominator of 5 and 6 is 30. Multiply every term of the equation by 30:
\[ 30 \cdot 2 + 30 \cdot \frac{x^2}{5} = 30 \cdot \frac{x}{6}. \]
This simplifies to:
\[ 60 + 6x^2 = 5x. \]
Next, we rearrange the equation into standard quadratic form:
\[ 6x^2 - 5x + 60 = 0. \]
Now, we apply the quadratic formula, given by
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]
where in our case \(a = 6\), \(b = -5\), and \(c = 60\). First, we calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = (-5)^2 - 4(6)(60) = 25 - 1440 = -1415. \]
Since the discriminant is negative, the roots will be complex. Now we substitute into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-1415}}{2(6)} = \frac{5 \pm \sqrt{1415}i}{12}. \]
Thus, simplifying the expression, we have:
\[ x = \frac{5}{12} \pm \frac{\sqrt{1415}}{12} i. \]
Therefore, the roots of the equation in simplest form are:
\[ \frac{5}{12} \pm \frac{\sqrt{1415}}{12} i. \]
So the answer can be expressed as:
\[ \frac{5}{12} \pm \frac{\sqrt{1415}}{12} i. \]
Thus, the answer is:
\[ \frac{5}{12} \pm \frac{\sqrt{1415}}{12} i. \]
Final answer:
The roots of the equation in simplest form are \(\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i\).
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