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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 = 28x (-5 + 28x + 12x2) = 120 Solving -5 + 28x + 12x2 = 120 Solving for variable 'x'. Reorder the terms: -5 + -120 + 28x + 12x2 = 120 + -120 Combine like terms: -5 + -120 = -125 -125 + 28x + 12x2 = 120 + -120 Combine like terms: 120 + -120 = 0 -125 + 28x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -10.41666667 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '10.41666667' to each side of the equation. -10.41666667 + 2.333333333x + 10.41666667 + x2 = 0 + 10.41666667 Reorder the terms: -10.41666667 + 10.41666667 + 2.333333333x + x2 = 0 + 10.41666667 Combine like terms: -10.41666667 + 10.41666667 = 0.00000000 0.00000000 + 2.333333333x + x2 = 0 + 10.41666667 2.333333333x + x2 = 0 + 10.41666667 Combine like terms: 0 + 10.41666667 = 10.41666667 2.333333333x + x2 = 10.41666667 The x term is 2.333333333x. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333x + 1.361111112 + x2 = 10.41666667 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 10.41666667 + 1.361111112 Combine like terms: 10.41666667 + 1.361111112 = 11.777777782 1.361111112 + 2.333333333x + x2 = 11.777777782 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 11.777777782 Calculate the square root of the right side: 3.431876714 Break this problem into two subproblems by setting (x + 1.166666667) equal to 3.431876714 and -3.431876714.Subproblem 1
x + 1.166666667 = 3.431876714 Simplifying x + 1.166666667 = 3.431876714 Reorder the terms: 1.166666667 + x = 3.431876714 Solving 1.166666667 + x = 3.431876714 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = 3.431876714 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 3.431876714 + -1.166666667 x = 3.431876714 + -1.166666667 Combine like terms: 3.431876714 + -1.166666667 = 2.265210047 x = 2.265210047 Simplifying x = 2.265210047Subproblem 2
x + 1.166666667 = -3.431876714 Simplifying x + 1.166666667 = -3.431876714 Reorder the terms: 1.166666667 + x = -3.431876714 Solving 1.166666667 + x = -3.431876714 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = -3.431876714 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -3.431876714 + -1.166666667 x = -3.431876714 + -1.166666667 Combine like terms: -3.431876714 + -1.166666667 = -4.598543381 x = -4.598543381 Simplifying x = -4.598543381Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.265210047, -4.598543381}
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