QuestionDecember 25, 2025

- Solve. (Put a comma between your answers.) (x+3)/(x+5)-(2x)/(x-5)=(4x+45)/(x^2)-25

- Solve. (Put a comma between your answers.) (x+3)/(x+5)-(2x)/(x-5)=(4x+45)/(x^2)-25
- Solve. (Put a comma between your answers.) (x+3)/(x+5)-(2x)/(x-5)=(4x+45)/(x^2)-25

Solution
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Answer

-6, -10 Explanation 1. Find common denominator The common denominator is x^2 - 25 = (x+5)(x-5). 2. Rewrite each term with the common denominator \frac{x+3}{x+5} = \frac{(x+3)(x-5)}{x^2-25}, \frac{2x}{x-5} = \frac{2x(x+5)}{x^2-25}. 3. Combine all terms over the common denominator \frac{(x+3)(x-5) - 2x(x+5)}{x^2-25} = \frac{4x+45}{x^2-25}. 4. Set numerators equal and expand (x+3)(x-5) - 2x(x+5) = 4x + 45. x^2 - 5x + 3x - 15 - 2x^2 - 10x = 4x + 45. x^2 - 2x - 15 - 2x^2 - 10x = 4x + 45. -x^2 - 12x - 15 = 4x + 45. 5. Move all terms to one side -x^2 - 12x - 15 - 4x - 45 = 0. -x^2 - 16x - 60 = 0. x^2 + 16x + 60 = 0. 6. Factor or use quadratic formula x = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 1 \cdot 60}}{2}. x = \frac{-16 \pm \sqrt{256 - 240}}{2}. x = \frac{-16 \pm \sqrt{16}}{2}. x = \frac{-16 \pm 4}{2}. 7. Solve for x x_1 = \frac{-16 + 4}{2} = -6, x_2 = \frac{-16 - 4}{2} = -10.

Explanation

1. Find common denominator<br /> The common denominator is $x^2 - 25 = (x+5)(x-5)$.<br />2. Rewrite each term with the common denominator<br /> $\frac{x+3}{x+5} = \frac{(x+3)(x-5)}{x^2-25}$, $\frac{2x}{x-5} = \frac{2x(x+5)}{x^2-25}$.<br />3. Combine all terms over the common denominator<br /> $\frac{(x+3)(x-5) - 2x(x+5)}{x^2-25} = \frac{4x+45}{x^2-25}$.<br />4. Set numerators equal and expand<br /> $(x+3)(x-5) - 2x(x+5) = 4x + 45$.<br /> $x^2 - 5x + 3x - 15 - 2x^2 - 10x = 4x + 45$.<br /> $x^2 - 2x - 15 - 2x^2 - 10x = 4x + 45$.<br /> $-x^2 - 12x - 15 = 4x + 45$.<br />5. Move all terms to one side<br /> $-x^2 - 12x - 15 - 4x - 45 = 0$.<br /> $-x^2 - 16x - 60 = 0$.<br /> $x^2 + 16x + 60 = 0$.<br />6. Factor or use quadratic formula<br /> $x = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 1 \cdot 60}}{2}$.<br /> $x = \frac{-16 \pm \sqrt{256 - 240}}{2}$.<br /> $x = \frac{-16 \pm \sqrt{16}}{2}$.<br /> $x = \frac{-16 \pm 4}{2}$.<br />7. Solve for $x$<br /> $x_1 = \frac{-16 + 4}{2} = -6$, $x_2 = \frac{-16 - 4}{2} = -10$.
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