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Simplifying 6x2 + -34x + 90 = 180 Reorder the terms: 90 + -34x + 6x2 = 180 Solving 90 + -34x + 6x2 = 180 Solving for variable 'x'. Reorder the terms: 90 + -180 + -34x + 6x2 = 180 + -180 Combine like terms: 90 + -180 = -90 -90 + -34x + 6x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -90 + -34x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-45 + -17x + 3x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-45 + -17x + 3x2)' equal to zero and attempt to solve: Simplifying -45 + -17x + 3x2 = 0 Solving -45 + -17x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -15 + -5.666666667x + x2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + -5.666666667x + 15 + x2 = 0 + 15 Reorder the terms: -15 + 15 + -5.666666667x + x2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + -5.666666667x + x2 = 0 + 15 -5.666666667x + x2 = 0 + 15 Combine like terms: 0 + 15 = 15 -5.666666667x + x2 = 15 The x term is -5.666666667x. Take half its coefficient (-2.833333334). Square it (8.027777782) and add it to both sides. Add '8.027777782' to each side of the equation. -5.666666667x + 8.027777782 + x2 = 15 + 8.027777782 Reorder the terms: 8.027777782 + -5.666666667x + x2 = 15 + 8.027777782 Combine like terms: 15 + 8.027777782 = 23.027777782 8.027777782 + -5.666666667x + x2 = 23.027777782 Factor a perfect square on the left side: (x + -2.833333334)(x + -2.833333334) = 23.027777782 Calculate the square root of the right side: 4.798726683 Break this problem into two subproblems by setting (x + -2.833333334) equal to 4.798726683 and -4.798726683.Subproblem 1
x + -2.833333334 = 4.798726683 Simplifying x + -2.833333334 = 4.798726683 Reorder the terms: -2.833333334 + x = 4.798726683 Solving -2.833333334 + x = 4.798726683 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.833333334' to each side of the equation. -2.833333334 + 2.833333334 + x = 4.798726683 + 2.833333334 Combine like terms: -2.833333334 + 2.833333334 = 0.000000000 0.000000000 + x = 4.798726683 + 2.833333334 x = 4.798726683 + 2.833333334 Combine like terms: 4.798726683 + 2.833333334 = 7.632060017 x = 7.632060017 Simplifying x = 7.632060017Subproblem 2
x + -2.833333334 = -4.798726683 Simplifying x + -2.833333334 = -4.798726683 Reorder the terms: -2.833333334 + x = -4.798726683 Solving -2.833333334 + x = -4.798726683 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.833333334' to each side of the equation. -2.833333334 + 2.833333334 + x = -4.798726683 + 2.833333334 Combine like terms: -2.833333334 + 2.833333334 = 0.000000000 0.000000000 + x = -4.798726683 + 2.833333334 x = -4.798726683 + 2.833333334 Combine like terms: -4.798726683 + 2.833333334 = -1.965393349 x = -1.965393349 Simplifying x = -1.965393349Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.632060017, -1.965393349}Solution
x = {7.632060017, -1.965393349}
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