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Simplifying (3x + 5)(2x + 7) = 22 Reorder the terms: (5 + 3x)(2x + 7) = 22 Reorder the terms: (5 + 3x)(7 + 2x) = 22 Multiply (5 + 3x) * (7 + 2x) (5(7 + 2x) + 3x * (7 + 2x)) = 22 ((7 * 5 + 2x * 5) + 3x * (7 + 2x)) = 22 ((35 + 10x) + 3x * (7 + 2x)) = 22 (35 + 10x + (7 * 3x + 2x * 3x)) = 22 (35 + 10x + (21x + 6x2)) = 22 Combine like terms: 10x + 21x = 31x (35 + 31x + 6x2) = 22 Solving 35 + 31x + 6x2 = 22 Solving for variable 'x'. Reorder the terms: 35 + -22 + 31x + 6x2 = 22 + -22 Combine like terms: 35 + -22 = 13 13 + 31x + 6x2 = 22 + -22 Combine like terms: 22 + -22 = 0 13 + 31x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 2.166666667 + 5.166666667x + x2 = 0 Move the constant term to the right: Add '-2.166666667' to each side of the equation. 2.166666667 + 5.166666667x + -2.166666667 + x2 = 0 + -2.166666667 Reorder the terms: 2.166666667 + -2.166666667 + 5.166666667x + x2 = 0 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + 5.166666667x + x2 = 0 + -2.166666667 5.166666667x + x2 = 0 + -2.166666667 Combine like terms: 0 + -2.166666667 = -2.166666667 5.166666667x + x2 = -2.166666667 The x term is 5.166666667x. Take half its coefficient (2.583333334). Square it (6.673611115) and add it to both sides. Add '6.673611115' to each side of the equation. 5.166666667x + 6.673611115 + x2 = -2.166666667 + 6.673611115 Reorder the terms: 6.673611115 + 5.166666667x + x2 = -2.166666667 + 6.673611115 Combine like terms: -2.166666667 + 6.673611115 = 4.506944448 6.673611115 + 5.166666667x + x2 = 4.506944448 Factor a perfect square on the left side: (x + 2.583333334)(x + 2.583333334) = 4.506944448 Calculate the square root of the right side: 2.122956535 Break this problem into two subproblems by setting (x + 2.583333334) equal to 2.122956535 and -2.122956535.Subproblem 1
x + 2.583333334 = 2.122956535 Simplifying x + 2.583333334 = 2.122956535 Reorder the terms: 2.583333334 + x = 2.122956535 Solving 2.583333334 + x = 2.122956535 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.583333334' to each side of the equation. 2.583333334 + -2.583333334 + x = 2.122956535 + -2.583333334 Combine like terms: 2.583333334 + -2.583333334 = 0.000000000 0.000000000 + x = 2.122956535 + -2.583333334 x = 2.122956535 + -2.583333334 Combine like terms: 2.122956535 + -2.583333334 = -0.460376799 x = -0.460376799 Simplifying x = -0.460376799Subproblem 2
x + 2.583333334 = -2.122956535 Simplifying x + 2.583333334 = -2.122956535 Reorder the terms: 2.583333334 + x = -2.122956535 Solving 2.583333334 + x = -2.122956535 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.583333334' to each side of the equation. 2.583333334 + -2.583333334 + x = -2.122956535 + -2.583333334 Combine like terms: 2.583333334 + -2.583333334 = 0.000000000 0.000000000 + x = -2.122956535 + -2.583333334 x = -2.122956535 + -2.583333334 Combine like terms: -2.122956535 + -2.583333334 = -4.706289869 x = -4.706289869 Simplifying x = -4.706289869Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.460376799, -4.706289869}
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