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Simplifying (x + 6)(x + 5) * 10 = 1 Reorder the terms: (6 + x)(x + 5) * 10 = 1 Reorder the terms: (6 + x)(5 + x) * 10 = 1 Reorder the terms for easier multiplication: 10(6 + x)(5 + x) = 1 Multiply (6 + x) * (5 + x) 10(6(5 + x) + x(5 + x)) = 1 10((5 * 6 + x * 6) + x(5 + x)) = 1 10((30 + 6x) + x(5 + x)) = 1 10(30 + 6x + (5 * x + x * x)) = 1 10(30 + 6x + (5x + x2)) = 1 Combine like terms: 6x + 5x = 11x 10(30 + 11x + x2) = 1 (30 * 10 + 11x * 10 + x2 * 10) = 1 (300 + 110x + 10x2) = 1 Solving 300 + 110x + 10x2 = 1 Solving for variable 'x'. Reorder the terms: 300 + -1 + 110x + 10x2 = 1 + -1 Combine like terms: 300 + -1 = 299 299 + 110x + 10x2 = 1 + -1 Combine like terms: 1 + -1 = 0 299 + 110x + 10x2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 29.9 + 11x + x2 = 0 Move the constant term to the right: Add '-29.9' to each side of the equation. 29.9 + 11x + -29.9 + x2 = 0 + -29.9 Reorder the terms: 29.9 + -29.9 + 11x + x2 = 0 + -29.9 Combine like terms: 29.9 + -29.9 = 0.0 0.0 + 11x + x2 = 0 + -29.9 11x + x2 = 0 + -29.9 Combine like terms: 0 + -29.9 = -29.9 11x + x2 = -29.9 The x term is 11x. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11x + 30.25 + x2 = -29.9 + 30.25 Reorder the terms: 30.25 + 11x + x2 = -29.9 + 30.25 Combine like terms: -29.9 + 30.25 = 0.35 30.25 + 11x + x2 = 0.35 Factor a perfect square on the left side: (x + 5.5)(x + 5.5) = 0.35 Calculate the square root of the right side: 0.591607978 Break this problem into two subproblems by setting (x + 5.5) equal to 0.591607978 and -0.591607978.Subproblem 1
x + 5.5 = 0.591607978 Simplifying x + 5.5 = 0.591607978 Reorder the terms: 5.5 + x = 0.591607978 Solving 5.5 + x = 0.591607978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + x = 0.591607978 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + x = 0.591607978 + -5.5 x = 0.591607978 + -5.5 Combine like terms: 0.591607978 + -5.5 = -4.908392022 x = -4.908392022 Simplifying x = -4.908392022Subproblem 2
x + 5.5 = -0.591607978 Simplifying x + 5.5 = -0.591607978 Reorder the terms: 5.5 + x = -0.591607978 Solving 5.5 + x = -0.591607978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + x = -0.591607978 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + x = -0.591607978 + -5.5 x = -0.591607978 + -5.5 Combine like terms: -0.591607978 + -5.5 = -6.091607978 x = -6.091607978 Simplifying x = -6.091607978Solution
The solution to the problem is based on the solutions from the subproblems. x = {-4.908392022, -6.091607978}
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