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Simplifying 18x2 + 66x + 45 = 0 Reorder the terms: 45 + 66x + 18x2 = 0 Solving 45 + 66x + 18x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(15 + 22x + 6x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(15 + 22x + 6x2)' equal to zero and attempt to solve: Simplifying 15 + 22x + 6x2 = 0 Solving 15 + 22x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 2.5 + 3.666666667x + x2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 3.666666667x + -2.5 + x2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 3.666666667x + x2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 3.666666667x + x2 = 0 + -2.5 3.666666667x + x2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 3.666666667x + x2 = -2.5 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 = -2.5 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667x + x2 = -2.5 + 3.361111114 Combine like terms: -2.5 + 3.361111114 = 0.861111114 3.361111114 + 3.666666667x + x2 = 0.861111114 Factor a perfect square on the left side: (x + 1.833333334)(x + 1.833333334) = 0.861111114 Calculate the square root of the right side: 0.927960729 Break this problem into two subproblems by setting (x + 1.833333334) equal to 0.927960729 and -0.927960729.Subproblem 1
x + 1.833333334 = 0.927960729 Simplifying x + 1.833333334 = 0.927960729 Reorder the terms: 1.833333334 + x = 0.927960729 Solving 1.833333334 + x = 0.927960729 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 = 0.927960729 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 0.927960729 + -1.833333334 x = 0.927960729 + -1.833333334 Combine like terms: 0.927960729 + -1.833333334 = -0.905372605 x = -0.905372605 Simplifying x = -0.905372605Subproblem 2
x + 1.833333334 = -0.927960729 Simplifying x + 1.833333334 = -0.927960729 Reorder the terms: 1.833333334 + x = -0.927960729 Solving 1.833333334 + x = -0.927960729 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 = -0.927960729 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -0.927960729 + -1.833333334 x = -0.927960729 + -1.833333334 Combine like terms: -0.927960729 + -1.833333334 = -2.761294063 x = -2.761294063 Simplifying x = -2.761294063Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.905372605, -2.761294063}Solution
x = {-0.905372605, -2.761294063}
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