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Simplifying (13 + x)(14 + x) = 0.9706 Multiply (13 + x) * (14 + x) (13(14 + x) + x(14 + x)) = 0.9706 ((14 * 13 + x * 13) + x(14 + x)) = 0.9706 ((182 + 13x) + x(14 + x)) = 0.9706 (182 + 13x + (14 * x + x * x)) = 0.9706 (182 + 13x + (14x + x2)) = 0.9706 Combine like terms: 13x + 14x = 27x (182 + 27x + x2) = 0.9706 Solving 182 + 27x + x2 = 0.9706 Solving for variable 'x'. Reorder the terms: 182 + -0.9706 + 27x + x2 = 0.9706 + -0.9706 Combine like terms: 182 + -0.9706 = 181.0294 181.0294 + 27x + x2 = 0.9706 + -0.9706 Combine like terms: 0.9706 + -0.9706 = 0.0000 181.0294 + 27x + x2 = 0.0000 Begin completing the square. Move the constant term to the right: Add '-181.0294' to each side of the equation. 181.0294 + 27x + -181.0294 + x2 = 0.0000 + -181.0294 Reorder the terms: 181.0294 + -181.0294 + 27x + x2 = 0.0000 + -181.0294 Combine like terms: 181.0294 + -181.0294 = 0.0000 0.0000 + 27x + x2 = 0.0000 + -181.0294 27x + x2 = 0.0000 + -181.0294 Combine like terms: 0.0000 + -181.0294 = -181.0294 27x + x2 = -181.0294 The x term is 27x. Take half its coefficient (13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. 27x + 182.25 + x2 = -181.0294 + 182.25 Reorder the terms: 182.25 + 27x + x2 = -181.0294 + 182.25 Combine like terms: -181.0294 + 182.25 = 1.2206 182.25 + 27x + x2 = 1.2206 Factor a perfect square on the left side: (x + 13.5)(x + 13.5) = 1.2206 Calculate the square root of the right side: 1.104807676 Break this problem into two subproblems by setting (x + 13.5) equal to 1.104807676 and -1.104807676.Subproblem 1
x + 13.5 = 1.104807676 Simplifying x + 13.5 = 1.104807676 Reorder the terms: 13.5 + x = 1.104807676 Solving 13.5 + x = 1.104807676 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + x = 1.104807676 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + x = 1.104807676 + -13.5 x = 1.104807676 + -13.5 Combine like terms: 1.104807676 + -13.5 = -12.395192324 x = -12.395192324 Simplifying x = -12.395192324Subproblem 2
x + 13.5 = -1.104807676 Simplifying x + 13.5 = -1.104807676 Reorder the terms: 13.5 + x = -1.104807676 Solving 13.5 + x = -1.104807676 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + x = -1.104807676 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + x = -1.104807676 + -13.5 x = -1.104807676 + -13.5 Combine like terms: -1.104807676 + -13.5 = -14.604807676 x = -14.604807676 Simplifying x = -14.604807676Solution
The solution to the problem is based on the solutions from the subproblems. x = {-12.395192324, -14.604807676}
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