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Simplifying 6n2 + 13n + -13 = 0 Reorder the terms: -13 + 13n + 6n2 = 0 Solving -13 + 13n + 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'. -2.166666667 + 2.166666667n + n2 = 0 Move the constant term to the right: Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667n + 2.166666667 + n2 = 0 + 2.166666667 Reorder the terms: -2.166666667 + 2.166666667 + 2.166666667n + n2 = 0 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + 2.166666667n + n2 = 0 + 2.166666667 2.166666667n + n2 = 0 + 2.166666667 Combine like terms: 0 + 2.166666667 = 2.166666667 2.166666667n + n2 = 2.166666667 The n term is 2.166666667n. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667n + 1.173611113 + n2 = 2.166666667 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667n + n2 = 2.166666667 + 1.173611113 Combine like terms: 2.166666667 + 1.173611113 = 3.34027778 1.173611113 + 2.166666667n + n2 = 3.34027778 Factor a perfect square on the left side: (n + 1.083333334)(n + 1.083333334) = 3.34027778 Calculate the square root of the right side: 1.827642684 Break this problem into two subproblems by setting (n + 1.083333334) equal to 1.827642684 and -1.827642684.Subproblem 1
n + 1.083333334 = 1.827642684 Simplifying n + 1.083333334 = 1.827642684 Reorder the terms: 1.083333334 + n = 1.827642684 Solving 1.083333334 + n = 1.827642684 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + n = 1.827642684 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + n = 1.827642684 + -1.083333334 n = 1.827642684 + -1.083333334 Combine like terms: 1.827642684 + -1.083333334 = 0.74430935 n = 0.74430935 Simplifying n = 0.74430935Subproblem 2
n + 1.083333334 = -1.827642684 Simplifying n + 1.083333334 = -1.827642684 Reorder the terms: 1.083333334 + n = -1.827642684 Solving 1.083333334 + n = -1.827642684 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + n = -1.827642684 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + n = -1.827642684 + -1.083333334 n = -1.827642684 + -1.083333334 Combine like terms: -1.827642684 + -1.083333334 = -2.910976018 n = -2.910976018 Simplifying n = -2.910976018Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.74430935, -2.910976018}
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