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Simplifying 3x2 + 10x + -27 = 0 Reorder the terms: -27 + 10x + 3x2 = 0 Solving -27 + 10x + 3x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -9 + 3.333333333x + x2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 3.333333333x + 9 + x2 = 0 + 9 Reorder the terms: -9 + 9 + 3.333333333x + x2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 3.333333333x + x2 = 0 + 9 3.333333333x + x2 = 0 + 9 Combine like terms: 0 + 9 = 9 3.333333333x + x2 = 9 The x term is 3.333333333x. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333x + 2.777777779 + x2 = 9 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333x + x2 = 9 + 2.777777779 Combine like terms: 9 + 2.777777779 = 11.777777779 2.777777779 + 3.333333333x + x2 = 11.777777779 Factor a perfect square on the left side: (x + 1.666666667)(x + 1.666666667) = 11.777777779 Calculate the square root of the right side: 3.431876714 Break this problem into two subproblems by setting (x + 1.666666667) equal to 3.431876714 and -3.431876714.Subproblem 1
x + 1.666666667 = 3.431876714 Simplifying x + 1.666666667 = 3.431876714 Reorder the terms: 1.666666667 + x = 3.431876714 Solving 1.666666667 + x = 3.431876714 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = 3.431876714 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = 3.431876714 + -1.666666667 x = 3.431876714 + -1.666666667 Combine like terms: 3.431876714 + -1.666666667 = 1.765210047 x = 1.765210047 Simplifying x = 1.765210047Subproblem 2
x + 1.666666667 = -3.431876714 Simplifying x + 1.666666667 = -3.431876714 Reorder the terms: 1.666666667 + x = -3.431876714 Solving 1.666666667 + x = -3.431876714 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = -3.431876714 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = -3.431876714 + -1.666666667 x = -3.431876714 + -1.666666667 Combine like terms: -3.431876714 + -1.666666667 = -5.098543381 x = -5.098543381 Simplifying x = -5.098543381Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.765210047, -5.098543381}
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