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Simplifying (x + 5)(3x + 4) = (2x + 1) + (x + 13) Reorder the terms: (5 + x)(3x + 4) = (2x + 1) + (x + 13) Reorder the terms: (5 + x)(4 + 3x) = (2x + 1) + (x + 13) Multiply (5 + x) * (4 + 3x) (5(4 + 3x) + x(4 + 3x)) = (2x + 1) + (x + 13) ((4 * 5 + 3x * 5) + x(4 + 3x)) = (2x + 1) + (x + 13) ((20 + 15x) + x(4 + 3x)) = (2x + 1) + (x + 13) (20 + 15x + (4 * x + 3x * x)) = (2x + 1) + (x + 13) (20 + 15x + (4x + 3x2)) = (2x + 1) + (x + 13) Combine like terms: 15x + 4x = 19x (20 + 19x + 3x2) = (2x + 1) + (x + 13) Reorder the terms: 20 + 19x + 3x2 = (1 + 2x) + (x + 13) Remove parenthesis around (1 + 2x) 20 + 19x + 3x2 = 1 + 2x + (x + 13) Reorder the terms: 20 + 19x + 3x2 = 1 + 2x + (13 + x) Remove parenthesis around (13 + x) 20 + 19x + 3x2 = 1 + 2x + 13 + x Reorder the terms: 20 + 19x + 3x2 = 1 + 13 + 2x + x Combine like terms: 1 + 13 = 14 20 + 19x + 3x2 = 14 + 2x + x Combine like terms: 2x + x = 3x 20 + 19x + 3x2 = 14 + 3x Solving 20 + 19x + 3x2 = 14 + 3x Solving for variable 'x'. Reorder the terms: 20 + -14 + 19x + -3x + 3x2 = 14 + 3x + -14 + -3x Combine like terms: 20 + -14 = 6 6 + 19x + -3x + 3x2 = 14 + 3x + -14 + -3x Combine like terms: 19x + -3x = 16x 6 + 16x + 3x2 = 14 + 3x + -14 + -3x Reorder the terms: 6 + 16x + 3x2 = 14 + -14 + 3x + -3x Combine like terms: 14 + -14 = 0 6 + 16x + 3x2 = 0 + 3x + -3x 6 + 16x + 3x2 = 3x + -3x Combine like terms: 3x + -3x = 0 6 + 16x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 2 + 5.333333333x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 5.333333333x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + 5.333333333x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 5.333333333x + x2 = 0 + -2 5.333333333x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 5.333333333x + x2 = -2 The x term is 5.333333333x. Take half its coefficient (2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. 5.333333333x + 7.111111113 + x2 = -2 + 7.111111113 Reorder the terms: 7.111111113 + 5.333333333x + x2 = -2 + 7.111111113 Combine like terms: -2 + 7.111111113 = 5.111111113 7.111111113 + 5.333333333x + x2 = 5.111111113 Factor a perfect square on the left side: (x + 2.666666667)(x + 2.666666667) = 5.111111113 Calculate the square root of the right side: 2.260776661 Break this problem into two subproblems by setting (x + 2.666666667) equal to 2.260776661 and -2.260776661.Subproblem 1
x + 2.666666667 = 2.260776661 Simplifying x + 2.666666667 = 2.260776661 Reorder the terms: 2.666666667 + x = 2.260776661 Solving 2.666666667 + x = 2.260776661 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + x = 2.260776661 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = 2.260776661 + -2.666666667 x = 2.260776661 + -2.666666667 Combine like terms: 2.260776661 + -2.666666667 = -0.405890006 x = -0.405890006 Simplifying x = -0.405890006Subproblem 2
x + 2.666666667 = -2.260776661 Simplifying x + 2.666666667 = -2.260776661 Reorder the terms: 2.666666667 + x = -2.260776661 Solving 2.666666667 + x = -2.260776661 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + x = -2.260776661 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = -2.260776661 + -2.666666667 x = -2.260776661 + -2.666666667 Combine like terms: -2.260776661 + -2.666666667 = -4.927443328 x = -4.927443328 Simplifying x = -4.927443328Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.405890006, -4.927443328}
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