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Simplifying 194x + -1(4 + -12x) + -2(41 + -1x) + 91x = 394x(5x + -222) 194x + (4 * -1 + -12x * -1) + -2(41 + -1x) + 91x = 394x(5x + -222) 194x + (-4 + 12x) + -2(41 + -1x) + 91x = 394x(5x + -222) 194x + -4 + 12x + (41 * -2 + -1x * -2) + 91x = 394x(5x + -222) 194x + -4 + 12x + (-82 + 2x) + 91x = 394x(5x + -222) Reorder the terms: -4 + -82 + 194x + 12x + 2x + 91x = 394x(5x + -222) Combine like terms: -4 + -82 = -86 -86 + 194x + 12x + 2x + 91x = 394x(5x + -222) Combine like terms: 194x + 12x = 206x -86 + 206x + 2x + 91x = 394x(5x + -222) Combine like terms: 206x + 2x = 208x -86 + 208x + 91x = 394x(5x + -222) Combine like terms: 208x + 91x = 299x -86 + 299x = 394x(5x + -222) Reorder the terms: -86 + 299x = 394x(-222 + 5x) -86 + 299x = (-222 * 394x + 5x * 394x) -86 + 299x = (-87468x + 1970x2) Solving -86 + 299x = -87468x + 1970x2 Solving for variable 'x'. Combine like terms: 299x + 87468x = 87767x -86 + 87767x + -1970x2 = -87468x + 1970x2 + 87468x + -1970x2 Reorder the terms: -86 + 87767x + -1970x2 = -87468x + 87468x + 1970x2 + -1970x2 Combine like terms: -87468x + 87468x = 0 -86 + 87767x + -1970x2 = 0 + 1970x2 + -1970x2 -86 + 87767x + -1970x2 = 1970x2 + -1970x2 Combine like terms: 1970x2 + -1970x2 = 0 -86 + 87767x + -1970x2 = 0 Begin completing the square. Divide all terms by -1970 the coefficient of the squared term: Divide each side by '-1970'. 0.04365482234 + -44.55177665x + x2 = 0 Move the constant term to the right: Add '-0.04365482234' to each side of the equation. 0.04365482234 + -44.55177665x + -0.04365482234 + x2 = 0 + -0.04365482234 Reorder the terms: 0.04365482234 + -0.04365482234 + -44.55177665x + x2 = 0 + -0.04365482234 Combine like terms: 0.04365482234 + -0.04365482234 = 0.00000000000 0.00000000000 + -44.55177665x + x2 = 0 + -0.04365482234 -44.55177665x + x2 = 0 + -0.04365482234 Combine like terms: 0 + -0.04365482234 = -0.04365482234 -44.55177665x + x2 = -0.04365482234 The x term is -44.55177665x. Take half its coefficient (-22.27588833). Square it (496.2152009) and add it to both sides. Add '496.2152009' to each side of the equation. -44.55177665x + 496.2152009 + x2 = -0.04365482234 + 496.2152009 Reorder the terms: 496.2152009 + -44.55177665x + x2 = -0.04365482234 + 496.2152009 Combine like terms: -0.04365482234 + 496.2152009 = 496.17154607766 496.2152009 + -44.55177665x + x2 = 496.17154607766 Factor a perfect square on the left side: (x + -22.27588833)(x + -22.27588833) = 496.17154607766 Calculate the square root of the right side: 22.274908442 Break this problem into two subproblems by setting (x + -22.27588833) equal to 22.274908442 and -22.274908442.Subproblem 1
x + -22.27588833 = 22.274908442 Simplifying x + -22.27588833 = 22.274908442 Reorder the terms: -22.27588833 + x = 22.274908442 Solving -22.27588833 + x = 22.274908442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '22.27588833' to each side of the equation. -22.27588833 + 22.27588833 + x = 22.274908442 + 22.27588833 Combine like terms: -22.27588833 + 22.27588833 = 0.00000000 0.00000000 + x = 22.274908442 + 22.27588833 x = 22.274908442 + 22.27588833 Combine like terms: 22.274908442 + 22.27588833 = 44.550796772 x = 44.550796772 Simplifying x = 44.550796772Subproblem 2
x + -22.27588833 = -22.274908442 Simplifying x + -22.27588833 = -22.274908442 Reorder the terms: -22.27588833 + x = -22.274908442 Solving -22.27588833 + x = -22.274908442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '22.27588833' to each side of the equation. -22.27588833 + 22.27588833 + x = -22.274908442 + 22.27588833 Combine like terms: -22.27588833 + 22.27588833 = 0.00000000 0.00000000 + x = -22.274908442 + 22.27588833 x = -22.274908442 + 22.27588833 Combine like terms: -22.274908442 + 22.27588833 = 0.000979888 x = 0.000979888 Simplifying x = 0.000979888Solution
The solution to the problem is based on the solutions from the subproblems. x = {44.550796772, 0.000979888}
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