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Simplifying 0.78(1 + -1x)(2 + -1x) = x Multiply (1 + -1x) * (2 + -1x) 0.78(1(2 + -1x) + -1x * (2 + -1x)) = x 0.78((2 * 1 + -1x * 1) + -1x * (2 + -1x)) = x 0.78((2 + -1x) + -1x * (2 + -1x)) = x 0.78(2 + -1x + (2 * -1x + -1x * -1x)) = x 0.78(2 + -1x + (-2x + 1x2)) = x Combine like terms: -1x + -2x = -3x 0.78(2 + -3x + 1x2) = x (2 * 0.78 + -3x * 0.78 + 1x2 * 0.78) = x (1.56 + -2.34x + 0.78x2) = x Solving 1.56 + -2.34x + 0.78x2 = x Solving for variable 'x'. Reorder the terms: 1.56 + -2.34x + -1x + 0.78x2 = x + -1x Combine like terms: -2.34x + -1x = -3.34x 1.56 + -3.34x + 0.78x2 = x + -1x Combine like terms: x + -1x = 0 1.56 + -3.34x + 0.78x2 = 0 Begin completing the square. Divide all terms by 0.78 the coefficient of the squared term: Divide each side by '0.78'. 2 + -4.282051282x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -4.282051282x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + -4.282051282x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -4.282051282x + x2 = 0 + -2 -4.282051282x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 -4.282051282x + x2 = -2 The x term is -4.282051282x. Take half its coefficient (-2.141025641). Square it (4.583990795) and add it to both sides. Add '4.583990795' to each side of the equation. -4.282051282x + 4.583990795 + x2 = -2 + 4.583990795 Reorder the terms: 4.583990795 + -4.282051282x + x2 = -2 + 4.583990795 Combine like terms: -2 + 4.583990795 = 2.583990795 4.583990795 + -4.282051282x + x2 = 2.583990795 Factor a perfect square on the left side: (x + -2.141025641)(x + -2.141025641) = 2.583990795 Calculate the square root of the right side: 1.607479641 Break this problem into two subproblems by setting (x + -2.141025641) equal to 1.607479641 and -1.607479641.Subproblem 1
x + -2.141025641 = 1.607479641 Simplifying x + -2.141025641 = 1.607479641 Reorder the terms: -2.141025641 + x = 1.607479641 Solving -2.141025641 + x = 1.607479641 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.141025641' to each side of the equation. -2.141025641 + 2.141025641 + x = 1.607479641 + 2.141025641 Combine like terms: -2.141025641 + 2.141025641 = 0.000000000 0.000000000 + x = 1.607479641 + 2.141025641 x = 1.607479641 + 2.141025641 Combine like terms: 1.607479641 + 2.141025641 = 3.748505282 x = 3.748505282 Simplifying x = 3.748505282Subproblem 2
x + -2.141025641 = -1.607479641 Simplifying x + -2.141025641 = -1.607479641 Reorder the terms: -2.141025641 + x = -1.607479641 Solving -2.141025641 + x = -1.607479641 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.141025641' to each side of the equation. -2.141025641 + 2.141025641 + x = -1.607479641 + 2.141025641 Combine like terms: -2.141025641 + 2.141025641 = 0.000000000 0.000000000 + x = -1.607479641 + 2.141025641 x = -1.607479641 + 2.141025641 Combine like terms: -1.607479641 + 2.141025641 = 0.533546 x = 0.533546 Simplifying x = 0.533546Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.748505282, 0.533546}
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