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Simplifying 90 + (11x)(6x + 5) = 180 Remove parenthesis around (11x) 90 + 11x(6x + 5) = 180 Reorder the terms: 90 + 11x(5 + 6x) = 180 90 + (5 * 11x + 6x * 11x) = 180 90 + (55x + 66x2) = 180 Solving 90 + 55x + 66x2 = 180 Solving for variable 'x'. Reorder the terms: 90 + -180 + 55x + 66x2 = 180 + -180 Combine like terms: 90 + -180 = -90 -90 + 55x + 66x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -90 + 55x + 66x2 = 0 Begin completing the square. Divide all terms by 66 the coefficient of the squared term: Divide each side by '66'. -1.363636364 + 0.8333333333x + x2 = 0 Move the constant term to the right: Add '1.363636364' to each side of the equation. -1.363636364 + 0.8333333333x + 1.363636364 + x2 = 0 + 1.363636364 Reorder the terms: -1.363636364 + 1.363636364 + 0.8333333333x + x2 = 0 + 1.363636364 Combine like terms: -1.363636364 + 1.363636364 = 0.000000000 0.000000000 + 0.8333333333x + x2 = 0 + 1.363636364 0.8333333333x + x2 = 0 + 1.363636364 Combine like terms: 0 + 1.363636364 = 1.363636364 0.8333333333x + x2 = 1.363636364 The x term is 0.8333333333x. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333x + 0.1736111111 + x2 = 1.363636364 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333x + x2 = 1.363636364 + 0.1736111111 Combine like terms: 1.363636364 + 0.1736111111 = 1.5372474751 0.1736111111 + 0.8333333333x + x2 = 1.5372474751 Factor a perfect square on the left side: (x + 0.4166666667)(x + 0.4166666667) = 1.5372474751 Calculate the square root of the right side: 1.239857845 Break this problem into two subproblems by setting (x + 0.4166666667) equal to 1.239857845 and -1.239857845.Subproblem 1
x + 0.4166666667 = 1.239857845 Simplifying x + 0.4166666667 = 1.239857845 Reorder the terms: 0.4166666667 + x = 1.239857845 Solving 0.4166666667 + x = 1.239857845 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 1.239857845 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 1.239857845 + -0.4166666667 x = 1.239857845 + -0.4166666667 Combine like terms: 1.239857845 + -0.4166666667 = 0.8231911783 x = 0.8231911783 Simplifying x = 0.8231911783Subproblem 2
x + 0.4166666667 = -1.239857845 Simplifying x + 0.4166666667 = -1.239857845 Reorder the terms: 0.4166666667 + x = -1.239857845 Solving 0.4166666667 + x = -1.239857845 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -1.239857845 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -1.239857845 + -0.4166666667 x = -1.239857845 + -0.4166666667 Combine like terms: -1.239857845 + -0.4166666667 = -1.6565245117 x = -1.6565245117 Simplifying x = -1.6565245117Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8231911783, -1.6565245117}
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