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Simplifying 90x2 + -120x + 31 = 0 Reorder the terms: 31 + -120x + 90x2 = 0 Solving 31 + -120x + 90x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 90 the coefficient of the squared term: Divide each side by '90'. 0.3444444444 + -1.333333333x + x2 = 0 Move the constant term to the right: Add '-0.3444444444' to each side of the equation. 0.3444444444 + -1.333333333x + -0.3444444444 + x2 = 0 + -0.3444444444 Reorder the terms: 0.3444444444 + -0.3444444444 + -1.333333333x + x2 = 0 + -0.3444444444 Combine like terms: 0.3444444444 + -0.3444444444 = 0.0000000000 0.0000000000 + -1.333333333x + x2 = 0 + -0.3444444444 -1.333333333x + x2 = 0 + -0.3444444444 Combine like terms: 0 + -0.3444444444 = -0.3444444444 -1.333333333x + x2 = -0.3444444444 The x term is -1.333333333x. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333x + 0.4444444442 + x2 = -0.3444444444 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333x + x2 = -0.3444444444 + 0.4444444442 Combine like terms: -0.3444444444 + 0.4444444442 = 0.0999999998 0.4444444442 + -1.333333333x + x2 = 0.0999999998 Factor a perfect square on the left side: (x + -0.6666666665)(x + -0.6666666665) = 0.0999999998 Calculate the square root of the right side: 0.316227766 Break this problem into two subproblems by setting (x + -0.6666666665) equal to 0.316227766 and -0.316227766.Subproblem 1
x + -0.6666666665 = 0.316227766 Simplifying x + -0.6666666665 = 0.316227766 Reorder the terms: -0.6666666665 + x = 0.316227766 Solving -0.6666666665 + x = 0.316227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = 0.316227766 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = 0.316227766 + 0.6666666665 x = 0.316227766 + 0.6666666665 Combine like terms: 0.316227766 + 0.6666666665 = 0.9828944325 x = 0.9828944325 Simplifying x = 0.9828944325Subproblem 2
x + -0.6666666665 = -0.316227766 Simplifying x + -0.6666666665 = -0.316227766 Reorder the terms: -0.6666666665 + x = -0.316227766 Solving -0.6666666665 + x = -0.316227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = -0.316227766 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = -0.316227766 + 0.6666666665 x = -0.316227766 + 0.6666666665 Combine like terms: -0.316227766 + 0.6666666665 = 0.3504389005 x = 0.3504389005 Simplifying x = 0.3504389005Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9828944325, 0.3504389005}
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