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Simplifying (3x + 5)(x + 7) = 90 Reorder the terms: (5 + 3x)(x + 7) = 90 Reorder the terms: (5 + 3x)(7 + x) = 90 Multiply (5 + 3x) * (7 + x) (5(7 + x) + 3x * (7 + x)) = 90 ((7 * 5 + x * 5) + 3x * (7 + x)) = 90 ((35 + 5x) + 3x * (7 + x)) = 90 (35 + 5x + (7 * 3x + x * 3x)) = 90 (35 + 5x + (21x + 3x2)) = 90 Combine like terms: 5x + 21x = 26x (35 + 26x + 3x2) = 90 Solving 35 + 26x + 3x2 = 90 Solving for variable 'x'. Reorder the terms: 35 + -90 + 26x + 3x2 = 90 + -90 Combine like terms: 35 + -90 = -55 -55 + 26x + 3x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -55 + 26x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -18.33333333 + 8.666666667x + x2 = 0 Move the constant term to the right: Add '18.33333333' to each side of the equation. -18.33333333 + 8.666666667x + 18.33333333 + x2 = 0 + 18.33333333 Reorder the terms: -18.33333333 + 18.33333333 + 8.666666667x + x2 = 0 + 18.33333333 Combine like terms: -18.33333333 + 18.33333333 = 0.00000000 0.00000000 + 8.666666667x + x2 = 0 + 18.33333333 8.666666667x + x2 = 0 + 18.33333333 Combine like terms: 0 + 18.33333333 = 18.33333333 8.666666667x + x2 = 18.33333333 The x term is 8.666666667x. Take half its coefficient (4.333333334). Square it (18.77777778) and add it to both sides. Add '18.77777778' to each side of the equation. 8.666666667x + 18.77777778 + x2 = 18.33333333 + 18.77777778 Reorder the terms: 18.77777778 + 8.666666667x + x2 = 18.33333333 + 18.77777778 Combine like terms: 18.33333333 + 18.77777778 = 37.11111111 18.77777778 + 8.666666667x + x2 = 37.11111111 Factor a perfect square on the left side: (x + 4.333333334)(x + 4.333333334) = 37.11111111 Calculate the square root of the right side: 6.091888961 Break this problem into two subproblems by setting (x + 4.333333334) equal to 6.091888961 and -6.091888961.Subproblem 1
x + 4.333333334 = 6.091888961 Simplifying x + 4.333333334 = 6.091888961 Reorder the terms: 4.333333334 + x = 6.091888961 Solving 4.333333334 + x = 6.091888961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.333333334' to each side of the equation. 4.333333334 + -4.333333334 + x = 6.091888961 + -4.333333334 Combine like terms: 4.333333334 + -4.333333334 = 0.000000000 0.000000000 + x = 6.091888961 + -4.333333334 x = 6.091888961 + -4.333333334 Combine like terms: 6.091888961 + -4.333333334 = 1.758555627 x = 1.758555627 Simplifying x = 1.758555627Subproblem 2
x + 4.333333334 = -6.091888961 Simplifying x + 4.333333334 = -6.091888961 Reorder the terms: 4.333333334 + x = -6.091888961 Solving 4.333333334 + x = -6.091888961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.333333334' to each side of the equation. 4.333333334 + -4.333333334 + x = -6.091888961 + -4.333333334 Combine like terms: 4.333333334 + -4.333333334 = 0.000000000 0.000000000 + x = -6.091888961 + -4.333333334 x = -6.091888961 + -4.333333334 Combine like terms: -6.091888961 + -4.333333334 = -10.425222295 x = -10.425222295 Simplifying x = -10.425222295Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.758555627, -10.425222295}
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