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Simplifying 6x2 + 19x + -6 = 180 Reorder the terms: -6 + 19x + 6x2 = 180 Solving -6 + 19x + 6x2 = 180 Solving for variable 'x'. Reorder the terms: -6 + -180 + 19x + 6x2 = 180 + -180 Combine like terms: -6 + -180 = -186 -186 + 19x + 6x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -186 + 19x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -31 + 3.166666667x + x2 = 0 Move the constant term to the right: Add '31' to each side of the equation. -31 + 3.166666667x + 31 + x2 = 0 + 31 Reorder the terms: -31 + 31 + 3.166666667x + x2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + 3.166666667x + x2 = 0 + 31 3.166666667x + x2 = 0 + 31 Combine like terms: 0 + 31 = 31 3.166666667x + x2 = 31 The x term is 3.166666667x. Take half its coefficient (1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. 3.166666667x + 2.506944447 + x2 = 31 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667x + x2 = 31 + 2.506944447 Combine like terms: 31 + 2.506944447 = 33.506944447 2.506944447 + 3.166666667x + x2 = 33.506944447 Factor a perfect square on the left side: (x + 1.583333334)(x + 1.583333334) = 33.506944447 Calculate the square root of the right side: 5.788518329 Break this problem into two subproblems by setting (x + 1.583333334) equal to 5.788518329 and -5.788518329.Subproblem 1
x + 1.583333334 = 5.788518329 Simplifying x + 1.583333334 = 5.788518329 Reorder the terms: 1.583333334 + x = 5.788518329 Solving 1.583333334 + x = 5.788518329 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + x = 5.788518329 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = 5.788518329 + -1.583333334 x = 5.788518329 + -1.583333334 Combine like terms: 5.788518329 + -1.583333334 = 4.205184995 x = 4.205184995 Simplifying x = 4.205184995Subproblem 2
x + 1.583333334 = -5.788518329 Simplifying x + 1.583333334 = -5.788518329 Reorder the terms: 1.583333334 + x = -5.788518329 Solving 1.583333334 + x = -5.788518329 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + x = -5.788518329 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = -5.788518329 + -1.583333334 x = -5.788518329 + -1.583333334 Combine like terms: -5.788518329 + -1.583333334 = -7.371851663 x = -7.371851663 Simplifying x = -7.371851663Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.205184995, -7.371851663}
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