4 (2)/(a-2)=(3a-1)/(2a+11)

To solve the equation \(4 \frac{2}{a-2} = \frac{3a-1}{2a+11}\), we need to follow these steps:<br /><br />1. **Rewrite the mixed number as an improper fraction:**<br /> \[<br /> 4 \frac{2}{a-2} = \frac{4(a-2) + 2}{a-2} = \frac{4a - 8 + 2}{a-2} = \frac{4a - 6}{a-2}<br /> \]<br /><br />2. **Set up the equation:**<br /> \[<br /> \frac{4a - 6}{a-2} = \frac{3a-1}{2a+11}<br /> \]<br /><br />3. **Cross-multiply to eliminate the fractions:**<br /> \[<br /> (4a - 6)(2a + 11) = (3a - 1)(a - 2)<br /> \]<br /><br />4. **Expand both sides:**<br /> \[<br /> (4a - 6)(2a + 11) = 8a^2 + 44a - 12a - 66 = 8a^2 + 32a - 66<br /> \]<br /> \[<br /> (3a - 1)(a - 2) = 3a^2 - 6a - a + 2 = 3a^2 - 7a + 2<br /> \]<br /><br />5. **Set the expanded forms equal to each other:**<br /> \[<br /> 8a^2 + 32a - 66 = 3a^2 - 7a + 2<br /> \]<br /><br />6. **Move all terms to one side to set the equation to zero:**<br /> \[<br /> 8a^2 + 32a - 66 - 3a^2 + 7a - 2 = 0<br /> \]<br /> \[<br /> 5a^2 + 39a - 68 = 0<br /> \]<br /><br />7. **Solve the quadratic equation using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):**<br /> Here, \(a = 5\), \(b = 39\), and \(c = -68\).<br /><br /> \[<br /> a = \frac{-39 \pm \sqrt{39^2 - 4 \cdot 5 \cdot (-68)}}{2 \cdot 5}<br /> \]<br /> \[<br /> a = \frac{-39 \pm \sqrt{1521 + 1360}}{10}<br /> \]<br /> \[<br /> a = \frac{-39 \pm \sqrt{2881}}{10}<br /> \]<br /><br />8. **Calculate the discriminant:**<br /> \[<br /> \sqrt{2881} \approx 53.7<br /> \]<br /><br />9. **Find the two possible solutions:**<br /> \[<br /> a = \frac{-39 + 53.7}{10} \approx 1.37<br /> \]<br /> \[<br /> a = \frac{-39 - 53.7}{10} \approx -9.27<br /> \]<br /><br />10. **Check for extraneous solutions:**<br /> Since the original equation involves division by \(a-2\), we must ensure that \(a \neq 2\). Both solutions \(a \approx 1.37\) and \(a \approx -9.27\) are valid as they do not make the denominator zero.<br /><br />Therefore, the solutions to the equation are:<br />\[<br />a \approx 1.37 \quad \text{or} \quad a \approx -9.27<br />\]