Answer and Solution please

To solve the equation 2x / (x + 1) - 3 / (x + 5) = -8x² / (x² + 6x + 5),we first need to simplify and solve it.1. Factor the denominator on the right side: The denominator x² + 6x + 5 can be factored as: (x + 1)(x + 5). Thus, the equation becomes: 2x / (x + 1) - 3 / (x + 5) = -8x² / [(x + 1)(x + 5)].2. Combine the left side with a common denominator: The common denominator for the left side is (x + 1)(x + 5). Rewrite each term with this common denominator: 2x / (x + 1) becomes (2x(x + 5)) / [(x + 1)(x + 5)], and -3 / (x + 5) becomes (-3(x + 1)) / [(x + 1)(x + 5)]. Combine these: (2x(x + 5) - 3(x + 1)) / [(x + 1)(x + 5)]. Simplify the numerator: 2x(x + 5) - 3(x + 1) = 2x² + 10x - 3x - 3 = 2x² + 7x - 3. So the left side of the equation becomes: (2x² + 7x - 3) / [(x + 1)(x + 5)]. Thus, the equation is: (2x² + 7x - 3) / [(x + 1)(x + 5)] = -8x² / [(x + 1)(x + 5)].3. Set the numerators equal to each other: Since the denominators are the same, equate the numerators: 2x² + 7x - 3 = -8x².4. Solve for x: Move all terms to one side of the equation: 2x² + 7x - 3 + 8x² = 0, 10x² + 7x - 3 = 0. Use the quadratic formula to solve: x = [-b ± sqrt(b² - 4ac)] / 2a, where a = 10, b = 7, and c = -3. Calculate the discriminant: b² - 4ac = 7² - 4(10)(-3), 49 + 120 = 169. The square root of 169 is 13. So: x = [-7 ± 13] / 20. This gives two solutions: x = [-7 + 13] / 20 = 6 / 20 = 3 / 10, x = [-7 - 13] / 20 = -20 / 20 = -1.The solutions are x = 3 / 10 and x = -1.

Answered by writerau19 | 2024-09-09