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Simplifying 0 = 3x2 + 11x + -1 Reorder the terms: 0 = -1 + 11x + 3x2 Solving 0 = -1 + 11x + 3x2 Solving for variable 'x'. Combine like terms: 0 + 1 = 1 1 + -11x + -3x2 = -1 + 11x + 3x2 + 1 + -11x + -3x2 Reorder the terms: 1 + -11x + -3x2 = -1 + 1 + 11x + -11x + 3x2 + -3x2 Combine like terms: -1 + 1 = 0 1 + -11x + -3x2 = 0 + 11x + -11x + 3x2 + -3x2 1 + -11x + -3x2 = 11x + -11x + 3x2 + -3x2 Combine like terms: 11x + -11x = 0 1 + -11x + -3x2 = 0 + 3x2 + -3x2 1 + -11x + -3x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 1 + -11x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -0.3333333333 + 3.666666667x + x2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 3.666666667x + 0.3333333333 + x2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 3.666666667x + x2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 3.666666667x + x2 = 0 + 0.3333333333 3.666666667x + x2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 3.666666667x + x2 = 0.3333333333 The x term is 3.666666667x. Take half its coefficient (1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. 3.666666667x + 3.361111114 + x2 = 0.3333333333 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667x + x2 = 0.3333333333 + 3.361111114 Combine like terms: 0.3333333333 + 3.361111114 = 3.6944444473 3.361111114 + 3.666666667x + x2 = 3.6944444473 Factor a perfect square on the left side: (x + 1.833333334)(x + 1.833333334) = 3.6944444473 Calculate the square root of the right side: 1.922093767 Break this problem into two subproblems by setting (x + 1.833333334) equal to 1.922093767 and -1.922093767.Subproblem 1
x + 1.833333334 = 1.922093767 Simplifying x + 1.833333334 = 1.922093767 Reorder the terms: 1.833333334 + x = 1.922093767 Solving 1.833333334 + x = 1.922093767 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = 1.922093767 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 1.922093767 + -1.833333334 x = 1.922093767 + -1.833333334 Combine like terms: 1.922093767 + -1.833333334 = 0.088760433 x = 0.088760433 Simplifying x = 0.088760433Subproblem 2
x + 1.833333334 = -1.922093767 Simplifying x + 1.833333334 = -1.922093767 Reorder the terms: 1.833333334 + x = -1.922093767 Solving 1.833333334 + x = -1.922093767 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = -1.922093767 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -1.922093767 + -1.833333334 x = -1.922093767 + -1.833333334 Combine like terms: -1.922093767 + -1.833333334 = -3.755427101 x = -3.755427101 Simplifying x = -3.755427101Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.088760433, -3.755427101}
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