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Simplifying (x + 40)(3x + -10) = 180 Reorder the terms: (40 + x)(3x + -10) = 180 Reorder the terms: (40 + x)(-10 + 3x) = 180 Multiply (40 + x) * (-10 + 3x) (40(-10 + 3x) + x(-10 + 3x)) = 180 ((-10 * 40 + 3x * 40) + x(-10 + 3x)) = 180 ((-400 + 120x) + x(-10 + 3x)) = 180 (-400 + 120x + (-10 * x + 3x * x)) = 180 (-400 + 120x + (-10x + 3x2)) = 180 Combine like terms: 120x + -10x = 110x (-400 + 110x + 3x2) = 180 Solving -400 + 110x + 3x2 = 180 Solving for variable 'x'. Reorder the terms: -400 + -180 + 110x + 3x2 = 180 + -180 Combine like terms: -400 + -180 = -580 -580 + 110x + 3x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -580 + 110x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -193.3333333 + 36.66666667x + x2 = 0 Move the constant term to the right: Add '193.3333333' to each side of the equation. -193.3333333 + 36.66666667x + 193.3333333 + x2 = 0 + 193.3333333 Reorder the terms: -193.3333333 + 193.3333333 + 36.66666667x + x2 = 0 + 193.3333333 Combine like terms: -193.3333333 + 193.3333333 = 0.0000000 0.0000000 + 36.66666667x + x2 = 0 + 193.3333333 36.66666667x + x2 = 0 + 193.3333333 Combine like terms: 0 + 193.3333333 = 193.3333333 36.66666667x + x2 = 193.3333333 The x term is 36.66666667x. Take half its coefficient (18.33333334). Square it (336.1111114) and add it to both sides. Add '336.1111114' to each side of the equation. 36.66666667x + 336.1111114 + x2 = 193.3333333 + 336.1111114 Reorder the terms: 336.1111114 + 36.66666667x + x2 = 193.3333333 + 336.1111114 Combine like terms: 193.3333333 + 336.1111114 = 529.4444447 336.1111114 + 36.66666667x + x2 = 529.4444447 Factor a perfect square on the left side: (x + 18.33333334)(x + 18.33333334) = 529.4444447 Calculate the square root of the right side: 23.009659813 Break this problem into two subproblems by setting (x + 18.33333334) equal to 23.009659813 and -23.009659813.Subproblem 1
x + 18.33333334 = 23.009659813 Simplifying x + 18.33333334 = 23.009659813 Reorder the terms: 18.33333334 + x = 23.009659813 Solving 18.33333334 + x = 23.009659813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-18.33333334' to each side of the equation. 18.33333334 + -18.33333334 + x = 23.009659813 + -18.33333334 Combine like terms: 18.33333334 + -18.33333334 = 0.00000000 0.00000000 + x = 23.009659813 + -18.33333334 x = 23.009659813 + -18.33333334 Combine like terms: 23.009659813 + -18.33333334 = 4.676326473 x = 4.676326473 Simplifying x = 4.676326473Subproblem 2
x + 18.33333334 = -23.009659813 Simplifying x + 18.33333334 = -23.009659813 Reorder the terms: 18.33333334 + x = -23.009659813 Solving 18.33333334 + x = -23.009659813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-18.33333334' to each side of the equation. 18.33333334 + -18.33333334 + x = -23.009659813 + -18.33333334 Combine like terms: 18.33333334 + -18.33333334 = 0.00000000 0.00000000 + x = -23.009659813 + -18.33333334 x = -23.009659813 + -18.33333334 Combine like terms: -23.009659813 + -18.33333334 = -41.342993153 x = -41.342993153 Simplifying x = -41.342993153Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.676326473, -41.342993153}
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