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Simplifying 6n2 + 19n + -15 = 0 Reorder the terms: -15 + 19n + 6n2 = 0 Solving -15 + 19n + 6n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.5 + 3.166666667n + n2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 3.166666667n + 2.5 + n2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 3.166666667n + n2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 3.166666667n + n2 = 0 + 2.5 3.166666667n + n2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 3.166666667n + n2 = 2.5 The n term is 3.166666667n. 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. 3.166666667n + 2.506944447 + n2 = 2.5 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667n + n2 = 2.5 + 2.506944447 Combine like terms: 2.5 + 2.506944447 = 5.006944447 2.506944447 + 3.166666667n + n2 = 5.006944447 Factor a perfect square on the left side: (n + 1.583333334)(n + 1.583333334) = 5.006944447 Calculate the square root of the right side: 2.237620264 Break this problem into two subproblems by setting (n + 1.583333334) equal to 2.237620264 and -2.237620264.Subproblem 1
n + 1.583333334 = 2.237620264 Simplifying n + 1.583333334 = 2.237620264 Reorder the terms: 1.583333334 + n = 2.237620264 Solving 1.583333334 + n = 2.237620264 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + n = 2.237620264 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + n = 2.237620264 + -1.583333334 n = 2.237620264 + -1.583333334 Combine like terms: 2.237620264 + -1.583333334 = 0.65428693 n = 0.65428693 Simplifying n = 0.65428693Subproblem 2
n + 1.583333334 = -2.237620264 Simplifying n + 1.583333334 = -2.237620264 Reorder the terms: 1.583333334 + n = -2.237620264 Solving 1.583333334 + n = -2.237620264 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + n = -2.237620264 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + n = -2.237620264 + -1.583333334 n = -2.237620264 + -1.583333334 Combine like terms: -2.237620264 + -1.583333334 = -3.820953598 n = -3.820953598 Simplifying n = -3.820953598Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.65428693, -3.820953598}
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