Questions: 1/(x+2) + 24/(x+3) = 13

Solution Steps

We start with the equation: \[ \frac{1}{x+2} + \frac{24}{x+3} = 13 \]

Multiply both sides by the common denominator \((x+2)(x+3)\): \[ (x+3) + 24(x+2) = 13(x+2)(x+3) \]

Expanding both sides gives: \[ x + 3 + 24x + 48 = 13(x^2 + 5x + 6) \] This simplifies to: \[ 25x + 51 = 13x^2 + 65x + 78 \]

Rearranging the equation results in: \[ 13x^2 + 40x + 27 = 0 \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 13\), \(b = 40\), and \(c = 27\), we find the solutions: \[ x = \frac{-40 \pm \sqrt{40^2 - 4 \cdot 13 \cdot 27}}{2 \cdot 13} \]

The discriminant is calculated as: \[ 40^2 - 4 \cdot 13 \cdot 27 = 1600 - 1404 = 196 \] Thus, the roots are: \[ x = \frac{-40 \pm 14}{26} \] This results in two solutions: \[ x = \frac{-27}{13} \quad \text{and} \quad x = -1 \]

Final Answer