X/2+3x - 5/3x-2 = 15x+10/4-9x^2
To solve the equation
\[ \frac{x}{2} + 3x - \frac{5}{3}x - 2 = \frac{15x + 10}{4 - 9x^2}, \]
we first simplify both sides.
Simplify the left side:
Combine the terms with \(x\):
\[ \frac{x}{2} + 3x - \frac{5}{3}x - 2 = x\left(\frac{1}{2} + 3 - \frac{5}{3}\right) - 2. \]
To combine the coefficients of \(x\), first express them with a common denominator:
\[ \frac{1}{2} = \frac{3}{6}, \quad 3 = \frac{18}{6}, \quad \frac{5}{3} = \frac{10}{6}. \]
So,
\[ \frac{1}{2} + 3 - \frac{5}{3} = \frac{3}{6} + \frac{18}{6} - \frac{10}{6} = \frac{11}{6}. \]
Therefore, the left side simplifies to:
\[ \frac{11}{6}x - 2. \]
Simplify the right side:
The right side is
\[ \frac{15x + 10}{4 - 9x^2}. \]
Set the equation:
Now we have the simplified equation:
\[ \frac{11}{6}x - 2 = \frac{15x + 10}{4 - 9x^2}. \]
To eliminate the fraction, cross-multiply:
\[ (11x - 12)(4 - 9x^2) = 6(15x + 10). \]
Expand both sides:
Left side: \[ 44 - 99x^2 - 48x + 108x^3 = 108x^3 - 99x^2 - 48x + 44. \]
Right side: \[ 90x + 60. \]
So, we equate:
\[ 108x^3 - 99x^2 - 48x + 44 = 90x + 60. \]
Rearrange the equation:
Move all terms to one side:
\[ 108x^3 - 99x^2 - 48x - 90x + 44 - 60 = 0, \]
simplifying to:
\[ 108x^3 - 99x^2 - 138x - 16 = 0. \]
Now we have a polynomial equation.
You can use numerical methods (like synthetic division or a graphing calculator) or factorization if possible to find the roots of the equation \(108x^3 - 99x^2 - 138x - 16 = 0\).
Alternatively, for practical purposes, using a calculator or software might provide you with the roots directly if this cubic equation is not easily factorizable.
