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Simplifying 3x4 + 11x3 + 4x2 = 0 Reorder the terms: 4x2 + 11x3 + 3x4 = 0 Solving 4x2 + 11x3 + 3x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(4 + 11x + 3x2) = 0Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(4 + 11x + 3x2)' equal to zero and attempt to solve: Simplifying 4 + 11x + 3x2 = 0 Solving 4 + 11x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + 3.666666667x + x2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 3.666666667x + -1.333333333 + x2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 3.666666667x + x2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 3.666666667x + x2 = 0 + -1.333333333 3.666666667x + x2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 3.666666667x + x2 = -1.333333333 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 = -1.333333333 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667x + x2 = -1.333333333 + 3.361111114 Combine like terms: -1.333333333 + 3.361111114 = 2.027777781 3.361111114 + 3.666666667x + x2 = 2.027777781 Factor a perfect square on the left side: (x + 1.833333334)(x + 1.833333334) = 2.027777781 Calculate the square root of the right side: 1.424000625 Break this problem into two subproblems by setting (x + 1.833333334) equal to 1.424000625 and -1.424000625.Subproblem 1
x + 1.833333334 = 1.424000625 Simplifying x + 1.833333334 = 1.424000625 Reorder the terms: 1.833333334 + x = 1.424000625 Solving 1.833333334 + x = 1.424000625 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 = 1.424000625 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 1.424000625 + -1.833333334 x = 1.424000625 + -1.833333334 Combine like terms: 1.424000625 + -1.833333334 = -0.409332709 x = -0.409332709 Simplifying x = -0.409332709Subproblem 2
x + 1.833333334 = -1.424000625 Simplifying x + 1.833333334 = -1.424000625 Reorder the terms: 1.833333334 + x = -1.424000625 Solving 1.833333334 + x = -1.424000625 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 = -1.424000625 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -1.424000625 + -1.833333334 x = -1.424000625 + -1.833333334 Combine like terms: -1.424000625 + -1.833333334 = -3.257333959 x = -3.257333959 Simplifying x = -3.257333959Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.409332709, -3.257333959}Solution
x = {0, -0.409332709, -3.257333959}
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