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Simplifying (5x + 7)(3x + 11) = 180 Reorder the terms: (7 + 5x)(3x + 11) = 180 Reorder the terms: (7 + 5x)(11 + 3x) = 180 Multiply (7 + 5x) * (11 + 3x) (7(11 + 3x) + 5x * (11 + 3x)) = 180 ((11 * 7 + 3x * 7) + 5x * (11 + 3x)) = 180 ((77 + 21x) + 5x * (11 + 3x)) = 180 (77 + 21x + (11 * 5x + 3x * 5x)) = 180 (77 + 21x + (55x + 15x2)) = 180 Combine like terms: 21x + 55x = 76x (77 + 76x + 15x2) = 180 Solving 77 + 76x + 15x2 = 180 Solving for variable 'x'. Reorder the terms: 77 + -180 + 76x + 15x2 = 180 + -180 Combine like terms: 77 + -180 = -103 -103 + 76x + 15x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -103 + 76x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -6.866666667 + 5.066666667x + x2 = 0 Move the constant term to the right: Add '6.866666667' to each side of the equation. -6.866666667 + 5.066666667x + 6.866666667 + x2 = 0 + 6.866666667 Reorder the terms: -6.866666667 + 6.866666667 + 5.066666667x + x2 = 0 + 6.866666667 Combine like terms: -6.866666667 + 6.866666667 = 0.000000000 0.000000000 + 5.066666667x + x2 = 0 + 6.866666667 5.066666667x + x2 = 0 + 6.866666667 Combine like terms: 0 + 6.866666667 = 6.866666667 5.066666667x + x2 = 6.866666667 The x term is 5.066666667x. Take half its coefficient (2.533333334). Square it (6.417777781) and add it to both sides. Add '6.417777781' to each side of the equation. 5.066666667x + 6.417777781 + x2 = 6.866666667 + 6.417777781 Reorder the terms: 6.417777781 + 5.066666667x + x2 = 6.866666667 + 6.417777781 Combine like terms: 6.866666667 + 6.417777781 = 13.284444448 6.417777781 + 5.066666667x + x2 = 13.284444448 Factor a perfect square on the left side: (x + 2.533333334)(x + 2.533333334) = 13.284444448 Calculate the square root of the right side: 3.644783183 Break this problem into two subproblems by setting (x + 2.533333334) equal to 3.644783183 and -3.644783183.Subproblem 1
x + 2.533333334 = 3.644783183 Simplifying x + 2.533333334 = 3.644783183 Reorder the terms: 2.533333334 + x = 3.644783183 Solving 2.533333334 + x = 3.644783183 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.533333334' to each side of the equation. 2.533333334 + -2.533333334 + x = 3.644783183 + -2.533333334 Combine like terms: 2.533333334 + -2.533333334 = 0.000000000 0.000000000 + x = 3.644783183 + -2.533333334 x = 3.644783183 + -2.533333334 Combine like terms: 3.644783183 + -2.533333334 = 1.111449849 x = 1.111449849 Simplifying x = 1.111449849Subproblem 2
x + 2.533333334 = -3.644783183 Simplifying x + 2.533333334 = -3.644783183 Reorder the terms: 2.533333334 + x = -3.644783183 Solving 2.533333334 + x = -3.644783183 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.533333334' to each side of the equation. 2.533333334 + -2.533333334 + x = -3.644783183 + -2.533333334 Combine like terms: 2.533333334 + -2.533333334 = 0.000000000 0.000000000 + x = -3.644783183 + -2.533333334 x = -3.644783183 + -2.533333334 Combine like terms: -3.644783183 + -2.533333334 = -6.178116517 x = -6.178116517 Simplifying x = -6.178116517Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.111449849, -6.178116517}
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