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Simplifying 3x2 + 11x + 9 = 0 Reorder the terms: 9 + 11x + 3x2 = 0 Solving 9 + 11x + 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'. 3 + 3.666666667x + x2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + 3.666666667x + -3 + x2 = 0 + -3 Reorder the terms: 3 + -3 + 3.666666667x + x2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 3.666666667x + x2 = 0 + -3 3.666666667x + x2 = 0 + -3 Combine like terms: 0 + -3 = -3 3.666666667x + x2 = -3 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 = -3 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667x + x2 = -3 + 3.361111114 Combine like terms: -3 + 3.361111114 = 0.361111114 3.361111114 + 3.666666667x + x2 = 0.361111114 Factor a perfect square on the left side: (x + 1.833333334)(x + 1.833333334) = 0.361111114 Calculate the square root of the right side: 0.600925215 Break this problem into two subproblems by setting (x + 1.833333334) equal to 0.600925215 and -0.600925215.Subproblem 1
x + 1.833333334 = 0.600925215 Simplifying x + 1.833333334 = 0.600925215 Reorder the terms: 1.833333334 + x = 0.600925215 Solving 1.833333334 + x = 0.600925215 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 = 0.600925215 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 0.600925215 + -1.833333334 x = 0.600925215 + -1.833333334 Combine like terms: 0.600925215 + -1.833333334 = -1.232408119 x = -1.232408119 Simplifying x = -1.232408119Subproblem 2
x + 1.833333334 = -0.600925215 Simplifying x + 1.833333334 = -0.600925215 Reorder the terms: 1.833333334 + x = -0.600925215 Solving 1.833333334 + x = -0.600925215 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 = -0.600925215 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -0.600925215 + -1.833333334 x = -0.600925215 + -1.833333334 Combine like terms: -0.600925215 + -1.833333334 = -2.434258549 x = -2.434258549 Simplifying x = -2.434258549Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.232408119, -2.434258549}
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