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Simplifying 6x2 + 16x + 9 = 0 Reorder the terms: 9 + 16x + 6x2 = 0 Solving 9 + 16x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 1.5 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 2.666666667x + -1.5 + x2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 2.666666667x + x2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 2.666666667x + x2 = 0 + -1.5 2.666666667x + x2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 2.666666667x + x2 = -1.5 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = -1.5 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = -1.5 + 1.777777780 Combine like terms: -1.5 + 1.777777780 = 0.27777778 1.777777780 + 2.666666667x + x2 = 0.27777778 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 0.27777778 Calculate the square root of the right side: 0.527046279 Break this problem into two subproblems by setting (x + 1.333333334) equal to 0.527046279 and -0.527046279.Subproblem 1
x + 1.333333334 = 0.527046279 Simplifying x + 1.333333334 = 0.527046279 Reorder the terms: 1.333333334 + x = 0.527046279 Solving 1.333333334 + x = 0.527046279 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 0.527046279 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 0.527046279 + -1.333333334 x = 0.527046279 + -1.333333334 Combine like terms: 0.527046279 + -1.333333334 = -0.806287055 x = -0.806287055 Simplifying x = -0.806287055Subproblem 2
x + 1.333333334 = -0.527046279 Simplifying x + 1.333333334 = -0.527046279 Reorder the terms: 1.333333334 + x = -0.527046279 Solving 1.333333334 + x = -0.527046279 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -0.527046279 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -0.527046279 + -1.333333334 x = -0.527046279 + -1.333333334 Combine like terms: -0.527046279 + -1.333333334 = -1.860379613 x = -1.860379613 Simplifying x = -1.860379613Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.806287055, -1.860379613}
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