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Simplifying 115x(1.00 + -1x)(2.00 + -1x) = 4x Multiply (1.00 + -1x) * (2.00 + -1x) 115x(1.00(2.00 + -1x) + -1x * (2.00 + -1x)) = 4x 115x((2.00 * 1.00 + -1x * 1.00) + -1x * (2.00 + -1x)) = 4x 115x((2 + -1x) + -1x * (2.00 + -1x)) = 4x 115x(2 + -1x + (2.00 * -1x + -1x * -1x)) = 4x 115x(2 + -1x + (-2x + 1x2)) = 4x Combine like terms: -1x + -2x = -3x 115x(2 + -3x + 1x2) = 4x (2 * 115x + -3x * 115x + 1x2 * 115x) = 4x (230x + -345x2 + 115x3) = 4x Solving 230x + -345x2 + 115x3 = 4x Solving for variable 'x'. Reorder the terms: 230x + -4x + -345x2 + 115x3 = 4x + -4x Combine like terms: 230x + -4x = 226x 226x + -345x2 + 115x3 = 4x + -4x Combine like terms: 4x + -4x = 0 226x + -345x2 + 115x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(226 + -345x + 115x2) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(226 + -345x + 115x2)' equal to zero and attempt to solve: Simplifying 226 + -345x + 115x2 = 0 Solving 226 + -345x + 115x2 = 0 Begin completing the square. Divide all terms by 115 the coefficient of the squared term: Divide each side by '115'. 1.965217391 + -3x + x2 = 0 Move the constant term to the right: Add '-1.965217391' to each side of the equation. 1.965217391 + -3x + -1.965217391 + x2 = 0 + -1.965217391 Reorder the terms: 1.965217391 + -1.965217391 + -3x + x2 = 0 + -1.965217391 Combine like terms: 1.965217391 + -1.965217391 = 0.000000000 0.000000000 + -3x + x2 = 0 + -1.965217391 -3x + x2 = 0 + -1.965217391 Combine like terms: 0 + -1.965217391 = -1.965217391 -3x + x2 = -1.965217391 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = -1.965217391 + 2.25 Reorder the terms: 2.25 + -3x + x2 = -1.965217391 + 2.25 Combine like terms: -1.965217391 + 2.25 = 0.284782609 2.25 + -3x + x2 = 0.284782609 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 0.284782609 Calculate the square root of the right side: 0.533650268 Break this problem into two subproblems by setting (x + -1.5) equal to 0.533650268 and -0.533650268.Subproblem 1
x + -1.5 = 0.533650268 Simplifying x + -1.5 = 0.533650268 Reorder the terms: -1.5 + x = 0.533650268 Solving -1.5 + x = 0.533650268 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 0.533650268 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.533650268 + 1.5 x = 0.533650268 + 1.5 Combine like terms: 0.533650268 + 1.5 = 2.033650268 x = 2.033650268 Simplifying x = 2.033650268Subproblem 2
x + -1.5 = -0.533650268 Simplifying x + -1.5 = -0.533650268 Reorder the terms: -1.5 + x = -0.533650268 Solving -1.5 + x = -0.533650268 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -0.533650268 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -0.533650268 + 1.5 x = -0.533650268 + 1.5 Combine like terms: -0.533650268 + 1.5 = 0.966349732 x = 0.966349732 Simplifying x = 0.966349732Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.033650268, 0.966349732}Solution
x = {0, 2.033650268, 0.966349732}
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