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Simplifying (1 + -1x)(2 + -1x)(115) = 2x2 Reorder the terms for easier multiplication: 115(1 + -1x)(2 + -1x) = 2x2 Multiply (1 + -1x) * (2 + -1x) 115(1(2 + -1x) + -1x * (2 + -1x)) = 2x2 115((2 * 1 + -1x * 1) + -1x * (2 + -1x)) = 2x2 115((2 + -1x) + -1x * (2 + -1x)) = 2x2 115(2 + -1x + (2 * -1x + -1x * -1x)) = 2x2 115(2 + -1x + (-2x + 1x2)) = 2x2 Combine like terms: -1x + -2x = -3x 115(2 + -3x + 1x2) = 2x2 (2 * 115 + -3x * 115 + 1x2 * 115) = 2x2 (230 + -345x + 115x2) = 2x2 Solving 230 + -345x + 115x2 = 2x2 Solving for variable 'x'. Combine like terms: 115x2 + -2x2 = 113x2 230 + -345x + 113x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 230 + -345x + 113x2 = 0 Begin completing the square. Divide all terms by 113 the coefficient of the squared term: Divide each side by '113'. 2.03539823 + -3.053097345x + x2 = 0 Move the constant term to the right: Add '-2.03539823' to each side of the equation. 2.03539823 + -3.053097345x + -2.03539823 + x2 = 0 + -2.03539823 Reorder the terms: 2.03539823 + -2.03539823 + -3.053097345x + x2 = 0 + -2.03539823 Combine like terms: 2.03539823 + -2.03539823 = 0.00000000 0.00000000 + -3.053097345x + x2 = 0 + -2.03539823 -3.053097345x + x2 = 0 + -2.03539823 Combine like terms: 0 + -2.03539823 = -2.03539823 -3.053097345x + x2 = -2.03539823 The x term is -3.053097345x. Take half its coefficient (-1.526548673). Square it (2.330350851) and add it to both sides. Add '2.330350851' to each side of the equation. -3.053097345x + 2.330350851 + x2 = -2.03539823 + 2.330350851 Reorder the terms: 2.330350851 + -3.053097345x + x2 = -2.03539823 + 2.330350851 Combine like terms: -2.03539823 + 2.330350851 = 0.294952621 2.330350851 + -3.053097345x + x2 = 0.294952621 Factor a perfect square on the left side: (x + -1.526548673)(x + -1.526548673) = 0.294952621 Calculate the square root of the right side: 0.543095407 Break this problem into two subproblems by setting (x + -1.526548673) equal to 0.543095407 and -0.543095407.Subproblem 1
x + -1.526548673 = 0.543095407 Simplifying x + -1.526548673 = 0.543095407 Reorder the terms: -1.526548673 + x = 0.543095407 Solving -1.526548673 + x = 0.543095407 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.526548673' to each side of the equation. -1.526548673 + 1.526548673 + x = 0.543095407 + 1.526548673 Combine like terms: -1.526548673 + 1.526548673 = 0.000000000 0.000000000 + x = 0.543095407 + 1.526548673 x = 0.543095407 + 1.526548673 Combine like terms: 0.543095407 + 1.526548673 = 2.06964408 x = 2.06964408 Simplifying x = 2.06964408Subproblem 2
x + -1.526548673 = -0.543095407 Simplifying x + -1.526548673 = -0.543095407 Reorder the terms: -1.526548673 + x = -0.543095407 Solving -1.526548673 + x = -0.543095407 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.526548673' to each side of the equation. -1.526548673 + 1.526548673 + x = -0.543095407 + 1.526548673 Combine like terms: -1.526548673 + 1.526548673 = 0.000000000 0.000000000 + x = -0.543095407 + 1.526548673 x = -0.543095407 + 1.526548673 Combine like terms: -0.543095407 + 1.526548673 = 0.983453266 x = 0.983453266 Simplifying x = 0.983453266Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.06964408, 0.983453266}
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