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Simplifying (5x + 29)(13x + -19) = 180 Reorder the terms: (29 + 5x)(13x + -19) = 180 Reorder the terms: (29 + 5x)(-19 + 13x) = 180 Multiply (29 + 5x) * (-19 + 13x) (29(-19 + 13x) + 5x * (-19 + 13x)) = 180 ((-19 * 29 + 13x * 29) + 5x * (-19 + 13x)) = 180 ((-551 + 377x) + 5x * (-19 + 13x)) = 180 (-551 + 377x + (-19 * 5x + 13x * 5x)) = 180 (-551 + 377x + (-95x + 65x2)) = 180 Combine like terms: 377x + -95x = 282x (-551 + 282x + 65x2) = 180 Solving -551 + 282x + 65x2 = 180 Solving for variable 'x'. Reorder the terms: -551 + -180 + 282x + 65x2 = 180 + -180 Combine like terms: -551 + -180 = -731 -731 + 282x + 65x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -731 + 282x + 65x2 = 0 Begin completing the square. Divide all terms by 65 the coefficient of the squared term: Divide each side by '65'. -11.24615385 + 4.338461538x + x2 = 0 Move the constant term to the right: Add '11.24615385' to each side of the equation. -11.24615385 + 4.338461538x + 11.24615385 + x2 = 0 + 11.24615385 Reorder the terms: -11.24615385 + 11.24615385 + 4.338461538x + x2 = 0 + 11.24615385 Combine like terms: -11.24615385 + 11.24615385 = 0.00000000 0.00000000 + 4.338461538x + x2 = 0 + 11.24615385 4.338461538x + x2 = 0 + 11.24615385 Combine like terms: 0 + 11.24615385 = 11.24615385 4.338461538x + x2 = 11.24615385 The x term is 4.338461538x. Take half its coefficient (2.169230769). Square it (4.705562129) and add it to both sides. Add '4.705562129' to each side of the equation. 4.338461538x + 4.705562129 + x2 = 11.24615385 + 4.705562129 Reorder the terms: 4.705562129 + 4.338461538x + x2 = 11.24615385 + 4.705562129 Combine like terms: 11.24615385 + 4.705562129 = 15.951715979 4.705562129 + 4.338461538x + x2 = 15.951715979 Factor a perfect square on the left side: (x + 2.169230769)(x + 2.169230769) = 15.951715979 Calculate the square root of the right side: 3.993959937 Break this problem into two subproblems by setting (x + 2.169230769) equal to 3.993959937 and -3.993959937.Subproblem 1
x + 2.169230769 = 3.993959937 Simplifying x + 2.169230769 = 3.993959937 Reorder the terms: 2.169230769 + x = 3.993959937 Solving 2.169230769 + x = 3.993959937 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.169230769' to each side of the equation. 2.169230769 + -2.169230769 + x = 3.993959937 + -2.169230769 Combine like terms: 2.169230769 + -2.169230769 = 0.000000000 0.000000000 + x = 3.993959937 + -2.169230769 x = 3.993959937 + -2.169230769 Combine like terms: 3.993959937 + -2.169230769 = 1.824729168 x = 1.824729168 Simplifying x = 1.824729168Subproblem 2
x + 2.169230769 = -3.993959937 Simplifying x + 2.169230769 = -3.993959937 Reorder the terms: 2.169230769 + x = -3.993959937 Solving 2.169230769 + x = -3.993959937 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.169230769' to each side of the equation. 2.169230769 + -2.169230769 + x = -3.993959937 + -2.169230769 Combine like terms: 2.169230769 + -2.169230769 = 0.000000000 0.000000000 + x = -3.993959937 + -2.169230769 x = -3.993959937 + -2.169230769 Combine like terms: -3.993959937 + -2.169230769 = -6.163190706 x = -6.163190706 Simplifying x = -6.163190706Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.824729168, -6.163190706}
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