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Simplifying 85 + 69 + (6x + 22)(x + 37) = 180 Reorder the terms: 85 + 69 + (22 + 6x)(x + 37) = 180 Reorder the terms: 85 + 69 + (22 + 6x)(37 + x) = 180 Multiply (22 + 6x) * (37 + x) 85 + 69 + (22(37 + x) + 6x * (37 + x)) = 180 85 + 69 + ((37 * 22 + x * 22) + 6x * (37 + x)) = 180 85 + 69 + ((814 + 22x) + 6x * (37 + x)) = 180 85 + 69 + (814 + 22x + (37 * 6x + x * 6x)) = 180 85 + 69 + (814 + 22x + (222x + 6x2)) = 180 Combine like terms: 22x + 222x = 244x 85 + 69 + (814 + 244x + 6x2) = 180 Combine like terms: 85 + 69 = 154 154 + 814 + 244x + 6x2 = 180 Combine like terms: 154 + 814 = 968 968 + 244x + 6x2 = 180 Solving 968 + 244x + 6x2 = 180 Solving for variable 'x'. Reorder the terms: 968 + -180 + 244x + 6x2 = 180 + -180 Combine like terms: 968 + -180 = 788 788 + 244x + 6x2 = 180 + -180 Combine like terms: 180 + -180 = 0 788 + 244x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(394 + 122x + 3x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(394 + 122x + 3x2)' equal to zero and attempt to solve: Simplifying 394 + 122x + 3x2 = 0 Solving 394 + 122x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 131.3333333 + 40.66666667x + x2 = 0 Move the constant term to the right: Add '-131.3333333' to each side of the equation. 131.3333333 + 40.66666667x + -131.3333333 + x2 = 0 + -131.3333333 Reorder the terms: 131.3333333 + -131.3333333 + 40.66666667x + x2 = 0 + -131.3333333 Combine like terms: 131.3333333 + -131.3333333 = 0.0000000 0.0000000 + 40.66666667x + x2 = 0 + -131.3333333 40.66666667x + x2 = 0 + -131.3333333 Combine like terms: 0 + -131.3333333 = -131.3333333 40.66666667x + x2 = -131.3333333 The x term is 40.66666667x. Take half its coefficient (20.33333334). Square it (413.4444447) and add it to both sides. Add '413.4444447' to each side of the equation. 40.66666667x + 413.4444447 + x2 = -131.3333333 + 413.4444447 Reorder the terms: 413.4444447 + 40.66666667x + x2 = -131.3333333 + 413.4444447 Combine like terms: -131.3333333 + 413.4444447 = 282.1111114 413.4444447 + 40.66666667x + x2 = 282.1111114 Factor a perfect square on the left side: (x + 20.33333334)(x + 20.33333334) = 282.1111114 Calculate the square root of the right side: 16.796163592 Break this problem into two subproblems by setting (x + 20.33333334) equal to 16.796163592 and -16.796163592.Subproblem 1
x + 20.33333334 = 16.796163592 Simplifying x + 20.33333334 = 16.796163592 Reorder the terms: 20.33333334 + x = 16.796163592 Solving 20.33333334 + x = 16.796163592 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-20.33333334' to each side of the equation. 20.33333334 + -20.33333334 + x = 16.796163592 + -20.33333334 Combine like terms: 20.33333334 + -20.33333334 = 0.00000000 0.00000000 + x = 16.796163592 + -20.33333334 x = 16.796163592 + -20.33333334 Combine like terms: 16.796163592 + -20.33333334 = -3.537169748 x = -3.537169748 Simplifying x = -3.537169748Subproblem 2
x + 20.33333334 = -16.796163592 Simplifying x + 20.33333334 = -16.796163592 Reorder the terms: 20.33333334 + x = -16.796163592 Solving 20.33333334 + x = -16.796163592 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-20.33333334' to each side of the equation. 20.33333334 + -20.33333334 + x = -16.796163592 + -20.33333334 Combine like terms: 20.33333334 + -20.33333334 = 0.00000000 0.00000000 + x = -16.796163592 + -20.33333334 x = -16.796163592 + -20.33333334 Combine like terms: -16.796163592 + -20.33333334 = -37.129496932 x = -37.129496932 Simplifying x = -37.129496932Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3.537169748, -37.129496932}Solution
x = {-3.537169748, -37.129496932}
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