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Simplifying 10n + 3(8 + 8n) = 6n(n + -4) 10n + (8 * 3 + 8n * 3) = 6n(n + -4) 10n + (24 + 24n) = 6n(n + -4) Reorder the terms: 24 + 10n + 24n = 6n(n + -4) Combine like terms: 10n + 24n = 34n 24 + 34n = 6n(n + -4) Reorder the terms: 24 + 34n = 6n(-4 + n) 24 + 34n = (-4 * 6n + n * 6n) 24 + 34n = (-24n + 6n2) Solving 24 + 34n = -24n + 6n2 Solving for variable 'n'. Combine like terms: 34n + 24n = 58n 24 + 58n + -6n2 = -24n + 6n2 + 24n + -6n2 Reorder the terms: 24 + 58n + -6n2 = -24n + 24n + 6n2 + -6n2 Combine like terms: -24n + 24n = 0 24 + 58n + -6n2 = 0 + 6n2 + -6n2 24 + 58n + -6n2 = 6n2 + -6n2 Combine like terms: 6n2 + -6n2 = 0 24 + 58n + -6n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(12 + 29n + -3n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(12 + 29n + -3n2)' equal to zero and attempt to solve: Simplifying 12 + 29n + -3n2 = 0 Solving 12 + 29n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -4 + -9.666666667n + n2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -9.666666667n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + -9.666666667n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -9.666666667n + n2 = 0 + 4 -9.666666667n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 -9.666666667n + n2 = 4 The n term is -9.666666667n. Take half its coefficient (-4.833333334). Square it (23.36111112) and add it to both sides. Add '23.36111112' to each side of the equation. -9.666666667n + 23.36111112 + n2 = 4 + 23.36111112 Reorder the terms: 23.36111112 + -9.666666667n + n2 = 4 + 23.36111112 Combine like terms: 4 + 23.36111112 = 27.36111112 23.36111112 + -9.666666667n + n2 = 27.36111112 Factor a perfect square on the left side: (n + -4.833333334)(n + -4.833333334) = 27.36111112 Calculate the square root of the right side: 5.230784943 Break this problem into two subproblems by setting (n + -4.833333334) equal to 5.230784943 and -5.230784943.Subproblem 1
n + -4.833333334 = 5.230784943 Simplifying n + -4.833333334 = 5.230784943 Reorder the terms: -4.833333334 + n = 5.230784943 Solving -4.833333334 + n = 5.230784943 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.833333334' to each side of the equation. -4.833333334 + 4.833333334 + n = 5.230784943 + 4.833333334 Combine like terms: -4.833333334 + 4.833333334 = 0.000000000 0.000000000 + n = 5.230784943 + 4.833333334 n = 5.230784943 + 4.833333334 Combine like terms: 5.230784943 + 4.833333334 = 10.064118277 n = 10.064118277 Simplifying n = 10.064118277Subproblem 2
n + -4.833333334 = -5.230784943 Simplifying n + -4.833333334 = -5.230784943 Reorder the terms: -4.833333334 + n = -5.230784943 Solving -4.833333334 + n = -5.230784943 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.833333334' to each side of the equation. -4.833333334 + 4.833333334 + n = -5.230784943 + 4.833333334 Combine like terms: -4.833333334 + 4.833333334 = 0.000000000 0.000000000 + n = -5.230784943 + 4.833333334 n = -5.230784943 + 4.833333334 Combine like terms: -5.230784943 + 4.833333334 = -0.397451609 n = -0.397451609 Simplifying n = -0.397451609Solution
The solution to the problem is based on the solutions from the subproblems. n = {10.064118277, -0.397451609}Solution
n = {10.064118277, -0.397451609}
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