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Simplifying (3 + 5x)(6x + 13) = 30 Reorder the terms: (3 + 5x)(13 + 6x) = 30 Multiply (3 + 5x) * (13 + 6x) (3(13 + 6x) + 5x * (13 + 6x)) = 30 ((13 * 3 + 6x * 3) + 5x * (13 + 6x)) = 30 ((39 + 18x) + 5x * (13 + 6x)) = 30 (39 + 18x + (13 * 5x + 6x * 5x)) = 30 (39 + 18x + (65x + 30x2)) = 30 Combine like terms: 18x + 65x = 83x (39 + 83x + 30x2) = 30 Solving 39 + 83x + 30x2 = 30 Solving for variable 'x'. Reorder the terms: 39 + -30 + 83x + 30x2 = 30 + -30 Combine like terms: 39 + -30 = 9 9 + 83x + 30x2 = 30 + -30 Combine like terms: 30 + -30 = 0 9 + 83x + 30x2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. 0.3 + 2.766666667x + x2 = 0.0 Move the constant term to the right: Add '-0.3' to each side of the equation. 0.3 + 2.766666667x + -0.3 + x2 = 0.0 + -0.3 Reorder the terms: 0.3 + -0.3 + 2.766666667x + x2 = 0.0 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + 2.766666667x + x2 = 0.0 + -0.3 2.766666667x + x2 = 0.0 + -0.3 Combine like terms: 0.0 + -0.3 = -0.3 2.766666667x + x2 = -0.3 The x term is 2.766666667x. Take half its coefficient (1.383333334). Square it (1.913611113) and add it to both sides. Add '1.913611113' to each side of the equation. 2.766666667x + 1.913611113 + x2 = -0.3 + 1.913611113 Reorder the terms: 1.913611113 + 2.766666667x + x2 = -0.3 + 1.913611113 Combine like terms: -0.3 + 1.913611113 = 1.613611113 1.913611113 + 2.766666667x + x2 = 1.613611113 Factor a perfect square on the left side: (x + 1.383333334)(x + 1.383333334) = 1.613611113 Calculate the square root of the right side: 1.270279935 Break this problem into two subproblems by setting (x + 1.383333334) equal to 1.270279935 and -1.270279935.Subproblem 1
x + 1.383333334 = 1.270279935 Simplifying x + 1.383333334 = 1.270279935 Reorder the terms: 1.383333334 + x = 1.270279935 Solving 1.383333334 + x = 1.270279935 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.383333334' to each side of the equation. 1.383333334 + -1.383333334 + x = 1.270279935 + -1.383333334 Combine like terms: 1.383333334 + -1.383333334 = 0.000000000 0.000000000 + x = 1.270279935 + -1.383333334 x = 1.270279935 + -1.383333334 Combine like terms: 1.270279935 + -1.383333334 = -0.113053399 x = -0.113053399 Simplifying x = -0.113053399Subproblem 2
x + 1.383333334 = -1.270279935 Simplifying x + 1.383333334 = -1.270279935 Reorder the terms: 1.383333334 + x = -1.270279935 Solving 1.383333334 + x = -1.270279935 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.383333334' to each side of the equation. 1.383333334 + -1.383333334 + x = -1.270279935 + -1.383333334 Combine like terms: 1.383333334 + -1.383333334 = 0.000000000 0.000000000 + x = -1.270279935 + -1.383333334 x = -1.270279935 + -1.383333334 Combine like terms: -1.270279935 + -1.383333334 = -2.653613269 x = -2.653613269 Simplifying x = -2.653613269Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.113053399, -2.653613269}
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