Thesolutionsfor theequation23 - x² - 2x = 0 are:x = 4x = -6Option A is the correct answer.We have,To solve therationalequation:1/(x - 1) - 1/(x + 3) - 1/5 = 0We need to find thecommondenominatorand combine the fractions on the left-hand side.The common denominator is (x - 1)(x + 3)(5).Multiplyingeach term by the common denominator:[(x + 3)(5) - (x - 1)(5) - (x - 1)(x + 3)] / [(x - 1)(x + 3)(5)] = 0Simplifying:[5(x + 3) - 5(x - 1) - (x - 1)(x + 3)] / [(x - 1)(x + 3)(5)] = 0[5x + 15 - 5x + 5 - (x² + 3x  - x - 3)] / [(x - 1)(x + 3)(5)] = 0[5x + 15 - 5x + 5 - x² - 3x + x + 3 ] / [(x - 1)(x + 3)(5)] = 0Simplifying further:[23 - x² - 2x ] / [(x - 1)(x + 3)(5)] = 0Now, we set thenumeratorequal tozerosince a fraction is zero if and only if its numerator is zero:23 - x² - 2x = 0To solve thequadraticequation23 - x² - 2x = 0, we can rearrange it to the standard form:x² + 2x - 23 = 0Now, we can solve it usingfactoring, completing the square, or the quadratic formula. Let's solve it using the quadratic formula:Thequadraticformulastates that for anequationin the formax² + bx + c = 0, thesolutionsfor x are given by:x = (-b ± √(b² - 4ac)) / (2a)For our equation x² + 2x - 23 = 0, the coefficients are a = 1, b = 2, andc = -23.Substitutingthese values into thequadraticformula:x = (-2 ± √(2² - 4(1)(-23))) / (2(1))Simplifying:x = (-2 ± √(4 + 92)) / 2x = (-2 ± √96) / 2x = (-2 ± 4√6) / 2Simplifying further:x = -1 ± 2√6Therefore,Thesolutionsfor theequation23 - x² - 2x = 0 are:x = -1 + 2√6 = 3.89 = 4x = -1 - 2√6 = -5.89 = -6Learn more aboutequationshere:brainly.com/question/17194269#SPJ6...

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