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Simplifying 10x(2x + -4) + 5 = 2x + -8 Reorder the terms: 10x(-4 + 2x) + 5 = 2x + -8 (-4 * 10x + 2x * 10x) + 5 = 2x + -8 (-40x + 20x2) + 5 = 2x + -8 Reorder the terms: 5 + -40x + 20x2 = 2x + -8 Reorder the terms: 5 + -40x + 20x2 = -8 + 2x Solving 5 + -40x + 20x2 = -8 + 2x Solving for variable 'x'. Reorder the terms: 5 + 8 + -40x + -2x + 20x2 = -8 + 2x + 8 + -2x Combine like terms: 5 + 8 = 13 13 + -40x + -2x + 20x2 = -8 + 2x + 8 + -2x Combine like terms: -40x + -2x = -42x 13 + -42x + 20x2 = -8 + 2x + 8 + -2x Reorder the terms: 13 + -42x + 20x2 = -8 + 8 + 2x + -2x Combine like terms: -8 + 8 = 0 13 + -42x + 20x2 = 0 + 2x + -2x 13 + -42x + 20x2 = 2x + -2x Combine like terms: 2x + -2x = 0 13 + -42x + 20x2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. 0.65 + -2.1x + x2 = 0.0 Move the constant term to the right: Add '-0.65' to each side of the equation. 0.65 + -2.1x + -0.65 + x2 = 0.0 + -0.65 Reorder the terms: 0.65 + -0.65 + -2.1x + x2 = 0.0 + -0.65 Combine like terms: 0.65 + -0.65 = 0.00 0.00 + -2.1x + x2 = 0.0 + -0.65 -2.1x + x2 = 0.0 + -0.65 Combine like terms: 0.0 + -0.65 = -0.65 -2.1x + x2 = -0.65 The x term is -2.1x. Take half its coefficient (-1.05). Square it (1.1025) and add it to both sides. Add '1.1025' to each side of the equation. -2.1x + 1.1025 + x2 = -0.65 + 1.1025 Reorder the terms: 1.1025 + -2.1x + x2 = -0.65 + 1.1025 Combine like terms: -0.65 + 1.1025 = 0.4525 1.1025 + -2.1x + x2 = 0.4525 Factor a perfect square on the left side: (x + -1.05)(x + -1.05) = 0.4525 Calculate the square root of the right side: 0.672681202 Break this problem into two subproblems by setting (x + -1.05) equal to 0.672681202 and -0.672681202.Subproblem 1
x + -1.05 = 0.672681202 Simplifying x + -1.05 = 0.672681202 Reorder the terms: -1.05 + x = 0.672681202 Solving -1.05 + x = 0.672681202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + x = 0.672681202 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + x = 0.672681202 + 1.05 x = 0.672681202 + 1.05 Combine like terms: 0.672681202 + 1.05 = 1.722681202 x = 1.722681202 Simplifying x = 1.722681202Subproblem 2
x + -1.05 = -0.672681202 Simplifying x + -1.05 = -0.672681202 Reorder the terms: -1.05 + x = -0.672681202 Solving -1.05 + x = -0.672681202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + x = -0.672681202 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + x = -0.672681202 + 1.05 x = -0.672681202 + 1.05 Combine like terms: -0.672681202 + 1.05 = 0.377318798 x = 0.377318798 Simplifying x = 0.377318798Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.722681202, 0.377318798}
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