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Simplifying 6x2 + 19x + 5 = 0 Reorder the terms: 5 + 19x + 6x2 = 0 Solving 5 + 19x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.8333333333 + 3.166666667x + x2 = 0 Move the constant term to the right: Add '-0.8333333333' to each side of the equation. 0.8333333333 + 3.166666667x + -0.8333333333 + x2 = 0 + -0.8333333333 Reorder the terms: 0.8333333333 + -0.8333333333 + 3.166666667x + x2 = 0 + -0.8333333333 Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000 0.0000000000 + 3.166666667x + x2 = 0 + -0.8333333333 3.166666667x + x2 = 0 + -0.8333333333 Combine like terms: 0 + -0.8333333333 = -0.8333333333 3.166666667x + x2 = -0.8333333333 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 = -0.8333333333 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667x + x2 = -0.8333333333 + 2.506944447 Combine like terms: -0.8333333333 + 2.506944447 = 1.6736111137 2.506944447 + 3.166666667x + x2 = 1.6736111137 Factor a perfect square on the left side: (x + 1.583333334)(x + 1.583333334) = 1.6736111137 Calculate the square root of the right side: 1.293681226 Break this problem into two subproblems by setting (x + 1.583333334) equal to 1.293681226 and -1.293681226.Subproblem 1
x + 1.583333334 = 1.293681226 Simplifying x + 1.583333334 = 1.293681226 Reorder the terms: 1.583333334 + x = 1.293681226 Solving 1.583333334 + x = 1.293681226 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 = 1.293681226 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = 1.293681226 + -1.583333334 x = 1.293681226 + -1.583333334 Combine like terms: 1.293681226 + -1.583333334 = -0.289652108 x = -0.289652108 Simplifying x = -0.289652108Subproblem 2
x + 1.583333334 = -1.293681226 Simplifying x + 1.583333334 = -1.293681226 Reorder the terms: 1.583333334 + x = -1.293681226 Solving 1.583333334 + x = -1.293681226 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 = -1.293681226 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = -1.293681226 + -1.583333334 x = -1.293681226 + -1.583333334 Combine like terms: -1.293681226 + -1.583333334 = -2.87701456 x = -2.87701456 Simplifying x = -2.87701456Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.289652108, -2.87701456}
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