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Simplifying (11x + -5)(7x + 27) = 180 Reorder the terms: (-5 + 11x)(7x + 27) = 180 Reorder the terms: (-5 + 11x)(27 + 7x) = 180 Multiply (-5 + 11x) * (27 + 7x) (-5(27 + 7x) + 11x * (27 + 7x)) = 180 ((27 * -5 + 7x * -5) + 11x * (27 + 7x)) = 180 ((-135 + -35x) + 11x * (27 + 7x)) = 180 (-135 + -35x + (27 * 11x + 7x * 11x)) = 180 (-135 + -35x + (297x + 77x2)) = 180 Combine like terms: -35x + 297x = 262x (-135 + 262x + 77x2) = 180 Solving -135 + 262x + 77x2 = 180 Solving for variable 'x'. Reorder the terms: -135 + -180 + 262x + 77x2 = 180 + -180 Combine like terms: -135 + -180 = -315 -315 + 262x + 77x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -315 + 262x + 77x2 = 0 Begin completing the square. Divide all terms by 77 the coefficient of the squared term: Divide each side by '77'. -4.090909091 + 3.402597403x + x2 = 0 Move the constant term to the right: Add '4.090909091' to each side of the equation. -4.090909091 + 3.402597403x + 4.090909091 + x2 = 0 + 4.090909091 Reorder the terms: -4.090909091 + 4.090909091 + 3.402597403x + x2 = 0 + 4.090909091 Combine like terms: -4.090909091 + 4.090909091 = 0.000000000 0.000000000 + 3.402597403x + x2 = 0 + 4.090909091 3.402597403x + x2 = 0 + 4.090909091 Combine like terms: 0 + 4.090909091 = 4.090909091 3.402597403x + x2 = 4.090909091 The x term is 3.402597403x. Take half its coefficient (1.701298702). Square it (2.894417273) and add it to both sides. Add '2.894417273' to each side of the equation. 3.402597403x + 2.894417273 + x2 = 4.090909091 + 2.894417273 Reorder the terms: 2.894417273 + 3.402597403x + x2 = 4.090909091 + 2.894417273 Combine like terms: 4.090909091 + 2.894417273 = 6.985326364 2.894417273 + 3.402597403x + x2 = 6.985326364 Factor a perfect square on the left side: (x + 1.701298702)(x + 1.701298702) = 6.985326364 Calculate the square root of the right side: 2.6429768 Break this problem into two subproblems by setting (x + 1.701298702) equal to 2.6429768 and -2.6429768.Subproblem 1
x + 1.701298702 = 2.6429768 Simplifying x + 1.701298702 = 2.6429768 Reorder the terms: 1.701298702 + x = 2.6429768 Solving 1.701298702 + x = 2.6429768 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.701298702' to each side of the equation. 1.701298702 + -1.701298702 + x = 2.6429768 + -1.701298702 Combine like terms: 1.701298702 + -1.701298702 = 0.000000000 0.000000000 + x = 2.6429768 + -1.701298702 x = 2.6429768 + -1.701298702 Combine like terms: 2.6429768 + -1.701298702 = 0.941678098 x = 0.941678098 Simplifying x = 0.941678098Subproblem 2
x + 1.701298702 = -2.6429768 Simplifying x + 1.701298702 = -2.6429768 Reorder the terms: 1.701298702 + x = -2.6429768 Solving 1.701298702 + x = -2.6429768 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.701298702' to each side of the equation. 1.701298702 + -1.701298702 + x = -2.6429768 + -1.701298702 Combine like terms: 1.701298702 + -1.701298702 = 0.000000000 0.000000000 + x = -2.6429768 + -1.701298702 x = -2.6429768 + -1.701298702 Combine like terms: -2.6429768 + -1.701298702 = -4.344275502 x = -4.344275502 Simplifying x = -4.344275502Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.941678098, -4.344275502}
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