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Simplifying x + x + 1.5x(3.5x + -8) = 90 Reorder the terms: x + x + 1.5x(-8 + 3.5x) = 90 x + x + (-8 * 1.5x + 3.5x * 1.5x) = 90 x + x + (-12x + 5.25x2) = 90 Combine like terms: x + x = 2x 2x + -12x + 5.25x2 = 90 Combine like terms: 2x + -12x = -10x -10x + 5.25x2 = 90 Solving -10x + 5.25x2 = 90 Solving for variable 'x'. Reorder the terms: -90 + -10x + 5.25x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + -10x + 5.25x2 = 0 Begin completing the square. Divide all terms by 5.25 the coefficient of the squared term: Divide each side by '5.25'. -17.14285714 + -1.904761905x + x2 = 0 Move the constant term to the right: Add '17.14285714' to each side of the equation. -17.14285714 + -1.904761905x + 17.14285714 + x2 = 0 + 17.14285714 Reorder the terms: -17.14285714 + 17.14285714 + -1.904761905x + x2 = 0 + 17.14285714 Combine like terms: -17.14285714 + 17.14285714 = 0.00000000 0.00000000 + -1.904761905x + x2 = 0 + 17.14285714 -1.904761905x + x2 = 0 + 17.14285714 Combine like terms: 0 + 17.14285714 = 17.14285714 -1.904761905x + x2 = 17.14285714 The x term is -1.904761905x. Take half its coefficient (-0.9523809525). Square it (0.9070294787) and add it to both sides. Add '0.9070294787' to each side of the equation. -1.904761905x + 0.9070294787 + x2 = 17.14285714 + 0.9070294787 Reorder the terms: 0.9070294787 + -1.904761905x + x2 = 17.14285714 + 0.9070294787 Combine like terms: 17.14285714 + 0.9070294787 = 18.0498866187 0.9070294787 + -1.904761905x + x2 = 18.0498866187 Factor a perfect square on the left side: (x + -0.9523809525)(x + -0.9523809525) = 18.0498866187 Calculate the square root of the right side: 4.248515814 Break this problem into two subproblems by setting (x + -0.9523809525) equal to 4.248515814 and -4.248515814.Subproblem 1
x + -0.9523809525 = 4.248515814 Simplifying x + -0.9523809525 = 4.248515814 Reorder the terms: -0.9523809525 + x = 4.248515814 Solving -0.9523809525 + x = 4.248515814 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9523809525' to each side of the equation. -0.9523809525 + 0.9523809525 + x = 4.248515814 + 0.9523809525 Combine like terms: -0.9523809525 + 0.9523809525 = 0.0000000000 0.0000000000 + x = 4.248515814 + 0.9523809525 x = 4.248515814 + 0.9523809525 Combine like terms: 4.248515814 + 0.9523809525 = 5.2008967665 x = 5.2008967665 Simplifying x = 5.2008967665Subproblem 2
x + -0.9523809525 = -4.248515814 Simplifying x + -0.9523809525 = -4.248515814 Reorder the terms: -0.9523809525 + x = -4.248515814 Solving -0.9523809525 + x = -4.248515814 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9523809525' to each side of the equation. -0.9523809525 + 0.9523809525 + x = -4.248515814 + 0.9523809525 Combine like terms: -0.9523809525 + 0.9523809525 = 0.0000000000 0.0000000000 + x = -4.248515814 + 0.9523809525 x = -4.248515814 + 0.9523809525 Combine like terms: -4.248515814 + 0.9523809525 = -3.2961348615 x = -3.2961348615 Simplifying x = -3.2961348615Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.2008967665, -3.2961348615}
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