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Simplifying -6n(6 + 2n) + -2(7n + 3) = 0 (6 * -6n + 2n * -6n) + -2(7n + 3) = 0 (-36n + -12n2) + -2(7n + 3) = 0 Reorder the terms: -36n + -12n2 + -2(3 + 7n) = 0 -36n + -12n2 + (3 * -2 + 7n * -2) = 0 -36n + -12n2 + (-6 + -14n) = 0 Reorder the terms: -6 + -36n + -14n + -12n2 = 0 Combine like terms: -36n + -14n = -50n -6 + -50n + -12n2 = 0 Solving -6 + -50n + -12n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '-2'. -2(3 + 25n + 6n2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(3 + 25n + 6n2)' equal to zero and attempt to solve: Simplifying 3 + 25n + 6n2 = 0 Solving 3 + 25n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.5 + 4.166666667n + n2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 4.166666667n + -0.5 + n2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 4.166666667n + n2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 4.166666667n + n2 = 0 + -0.5 4.166666667n + n2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 4.166666667n + n2 = -0.5 The n term is 4.166666667n. Take half its coefficient (2.083333334). Square it (4.340277781) and add it to both sides. Add '4.340277781' to each side of the equation. 4.166666667n + 4.340277781 + n2 = -0.5 + 4.340277781 Reorder the terms: 4.340277781 + 4.166666667n + n2 = -0.5 + 4.340277781 Combine like terms: -0.5 + 4.340277781 = 3.840277781 4.340277781 + 4.166666667n + n2 = 3.840277781 Factor a perfect square on the left side: (n + 2.083333334)(n + 2.083333334) = 3.840277781 Calculate the square root of the right side: 1.95966267 Break this problem into two subproblems by setting (n + 2.083333334) equal to 1.95966267 and -1.95966267.Subproblem 1
n + 2.083333334 = 1.95966267 Simplifying n + 2.083333334 = 1.95966267 Reorder the terms: 2.083333334 + n = 1.95966267 Solving 2.083333334 + n = 1.95966267 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.083333334' to each side of the equation. 2.083333334 + -2.083333334 + n = 1.95966267 + -2.083333334 Combine like terms: 2.083333334 + -2.083333334 = 0.000000000 0.000000000 + n = 1.95966267 + -2.083333334 n = 1.95966267 + -2.083333334 Combine like terms: 1.95966267 + -2.083333334 = -0.123670664 n = -0.123670664 Simplifying n = -0.123670664Subproblem 2
n + 2.083333334 = -1.95966267 Simplifying n + 2.083333334 = -1.95966267 Reorder the terms: 2.083333334 + n = -1.95966267 Solving 2.083333334 + n = -1.95966267 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.083333334' to each side of the equation. 2.083333334 + -2.083333334 + n = -1.95966267 + -2.083333334 Combine like terms: 2.083333334 + -2.083333334 = 0.000000000 0.000000000 + n = -1.95966267 + -2.083333334 n = -1.95966267 + -2.083333334 Combine like terms: -1.95966267 + -2.083333334 = -4.042996004 n = -4.042996004 Simplifying n = -4.042996004Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.123670664, -4.042996004}Solution
n = {-0.123670664, -4.042996004}
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