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Simplifying (x + 10)(3x + -5) + x = 180 Reorder the terms: (10 + x)(3x + -5) + x = 180 Reorder the terms: (10 + x)(-5 + 3x) + x = 180 Multiply (10 + x) * (-5 + 3x) (10(-5 + 3x) + x(-5 + 3x)) + x = 180 ((-5 * 10 + 3x * 10) + x(-5 + 3x)) + x = 180 ((-50 + 30x) + x(-5 + 3x)) + x = 180 (-50 + 30x + (-5 * x + 3x * x)) + x = 180 (-50 + 30x + (-5x + 3x2)) + x = 180 Combine like terms: 30x + -5x = 25x (-50 + 25x + 3x2) + x = 180 Reorder the terms: -50 + 25x + x + 3x2 = 180 Combine like terms: 25x + x = 26x -50 + 26x + 3x2 = 180 Solving -50 + 26x + 3x2 = 180 Solving for variable 'x'. Reorder the terms: -50 + -180 + 26x + 3x2 = 180 + -180 Combine like terms: -50 + -180 = -230 -230 + 26x + 3x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -230 + 26x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -76.66666667 + 8.666666667x + x2 = 0 Move the constant term to the right: Add '76.66666667' to each side of the equation. -76.66666667 + 8.666666667x + 76.66666667 + x2 = 0 + 76.66666667 Reorder the terms: -76.66666667 + 76.66666667 + 8.666666667x + x2 = 0 + 76.66666667 Combine like terms: -76.66666667 + 76.66666667 = 0.00000000 0.00000000 + 8.666666667x + x2 = 0 + 76.66666667 8.666666667x + x2 = 0 + 76.66666667 Combine like terms: 0 + 76.66666667 = 76.66666667 8.666666667x + x2 = 76.66666667 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 = 76.66666667 + 18.77777778 Reorder the terms: 18.77777778 + 8.666666667x + x2 = 76.66666667 + 18.77777778 Combine like terms: 76.66666667 + 18.77777778 = 95.44444445 18.77777778 + 8.666666667x + x2 = 95.44444445 Factor a perfect square on the left side: (x + 4.333333334)(x + 4.333333334) = 95.44444445 Calculate the square root of the right side: 9.76956726 Break this problem into two subproblems by setting (x + 4.333333334) equal to 9.76956726 and -9.76956726.Subproblem 1
x + 4.333333334 = 9.76956726 Simplifying x + 4.333333334 = 9.76956726 Reorder the terms: 4.333333334 + x = 9.76956726 Solving 4.333333334 + x = 9.76956726 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 = 9.76956726 + -4.333333334 Combine like terms: 4.333333334 + -4.333333334 = 0.000000000 0.000000000 + x = 9.76956726 + -4.333333334 x = 9.76956726 + -4.333333334 Combine like terms: 9.76956726 + -4.333333334 = 5.436233926 x = 5.436233926 Simplifying x = 5.436233926Subproblem 2
x + 4.333333334 = -9.76956726 Simplifying x + 4.333333334 = -9.76956726 Reorder the terms: 4.333333334 + x = -9.76956726 Solving 4.333333334 + x = -9.76956726 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 = -9.76956726 + -4.333333334 Combine like terms: 4.333333334 + -4.333333334 = 0.000000000 0.000000000 + x = -9.76956726 + -4.333333334 x = -9.76956726 + -4.333333334 Combine like terms: -9.76956726 + -4.333333334 = -14.102900594 x = -14.102900594 Simplifying x = -14.102900594Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.436233926, -14.102900594}
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