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Simplifying 0 = 36x2 + -168x + 120 Reorder the terms: 0 = 120 + -168x + 36x2 Solving 0 = 120 + -168x + 36x2 Solving for variable 'x'. Combine like terms: 0 + -120 = -120 -120 + 168x + -36x2 = 120 + -168x + 36x2 + -120 + 168x + -36x2 Reorder the terms: -120 + 168x + -36x2 = 120 + -120 + -168x + 168x + 36x2 + -36x2 Combine like terms: 120 + -120 = 0 -120 + 168x + -36x2 = 0 + -168x + 168x + 36x2 + -36x2 -120 + 168x + -36x2 = -168x + 168x + 36x2 + -36x2 Combine like terms: -168x + 168x = 0 -120 + 168x + -36x2 = 0 + 36x2 + -36x2 -120 + 168x + -36x2 = 36x2 + -36x2 Combine like terms: 36x2 + -36x2 = 0 -120 + 168x + -36x2 = 0 Factor out the Greatest Common Factor (GCF), '12'. 12(-10 + 14x + -3x2) = 0 Ignore the factor 12.Subproblem 1
Set the factor '(-10 + 14x + -3x2)' equal to zero and attempt to solve: Simplifying -10 + 14x + -3x2 = 0 Solving -10 + 14x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 3.333333333 + -4.666666667x + x2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + -4.666666667x + -3.333333333 + x2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + -4.666666667x + x2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + -4.666666667x + x2 = 0 + -3.333333333 -4.666666667x + x2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 -4.666666667x + x2 = -3.333333333 The x term is -4.666666667x. Take half its coefficient (-2.333333334). Square it (5.444444448) and add it to both sides. Add '5.444444448' to each side of the equation. -4.666666667x + 5.444444448 + x2 = -3.333333333 + 5.444444448 Reorder the terms: 5.444444448 + -4.666666667x + x2 = -3.333333333 + 5.444444448 Combine like terms: -3.333333333 + 5.444444448 = 2.111111115 5.444444448 + -4.666666667x + x2 = 2.111111115 Factor a perfect square on the left side: (x + -2.333333334)(x + -2.333333334) = 2.111111115 Calculate the square root of the right side: 1.452966316 Break this problem into two subproblems by setting (x + -2.333333334) equal to 1.452966316 and -1.452966316.Subproblem 1
x + -2.333333334 = 1.452966316 Simplifying x + -2.333333334 = 1.452966316 Reorder the terms: -2.333333334 + x = 1.452966316 Solving -2.333333334 + x = 1.452966316 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + x = 1.452966316 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + x = 1.452966316 + 2.333333334 x = 1.452966316 + 2.333333334 Combine like terms: 1.452966316 + 2.333333334 = 3.78629965 x = 3.78629965 Simplifying x = 3.78629965Subproblem 2
x + -2.333333334 = -1.452966316 Simplifying x + -2.333333334 = -1.452966316 Reorder the terms: -2.333333334 + x = -1.452966316 Solving -2.333333334 + x = -1.452966316 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + x = -1.452966316 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + x = -1.452966316 + 2.333333334 x = -1.452966316 + 2.333333334 Combine like terms: -1.452966316 + 2.333333334 = 0.880367018 x = 0.880367018 Simplifying x = 0.880367018Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.78629965, 0.880367018}Solution
x = {3.78629965, 0.880367018}
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