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Simplifying (2x + 5)(6x + 1) = 120 Reorder the terms: (5 + 2x)(6x + 1) = 120 Reorder the terms: (5 + 2x)(1 + 6x) = 120 Multiply (5 + 2x) * (1 + 6x) (5(1 + 6x) + 2x * (1 + 6x)) = 120 ((1 * 5 + 6x * 5) + 2x * (1 + 6x)) = 120 ((5 + 30x) + 2x * (1 + 6x)) = 120 (5 + 30x + (1 * 2x + 6x * 2x)) = 120 (5 + 30x + (2x + 12x2)) = 120 Combine like terms: 30x + 2x = 32x (5 + 32x + 12x2) = 120 Solving 5 + 32x + 12x2 = 120 Solving for variable 'x'. Reorder the terms: 5 + -120 + 32x + 12x2 = 120 + -120 Combine like terms: 5 + -120 = -115 -115 + 32x + 12x2 = 120 + -120 Combine like terms: 120 + -120 = 0 -115 + 32x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -9.583333333 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '9.583333333' to each side of the equation. -9.583333333 + 2.666666667x + 9.583333333 + x2 = 0 + 9.583333333 Reorder the terms: -9.583333333 + 9.583333333 + 2.666666667x + x2 = 0 + 9.583333333 Combine like terms: -9.583333333 + 9.583333333 = 0.000000000 0.000000000 + 2.666666667x + x2 = 0 + 9.583333333 2.666666667x + x2 = 0 + 9.583333333 Combine like terms: 0 + 9.583333333 = 9.583333333 2.666666667x + x2 = 9.583333333 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = 9.583333333 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 9.583333333 + 1.777777780 Combine like terms: 9.583333333 + 1.777777780 = 11.361111113 1.777777780 + 2.666666667x + x2 = 11.361111113 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 11.361111113 Calculate the square root of the right side: 3.370624736 Break this problem into two subproblems by setting (x + 1.333333334) equal to 3.370624736 and -3.370624736.Subproblem 1
x + 1.333333334 = 3.370624736 Simplifying x + 1.333333334 = 3.370624736 Reorder the terms: 1.333333334 + x = 3.370624736 Solving 1.333333334 + x = 3.370624736 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 3.370624736 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 3.370624736 + -1.333333334 x = 3.370624736 + -1.333333334 Combine like terms: 3.370624736 + -1.333333334 = 2.037291402 x = 2.037291402 Simplifying x = 2.037291402Subproblem 2
x + 1.333333334 = -3.370624736 Simplifying x + 1.333333334 = -3.370624736 Reorder the terms: 1.333333334 + x = -3.370624736 Solving 1.333333334 + x = -3.370624736 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -3.370624736 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -3.370624736 + -1.333333334 x = -3.370624736 + -1.333333334 Combine like terms: -3.370624736 + -1.333333334 = -4.70395807 x = -4.70395807 Simplifying x = -4.70395807Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.037291402, -4.70395807}
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