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Simplifying n(1 + 10n) + -1n(4n + -3) * n = 0 (1 * n + 10n * n) + -1n(4n + -3) * n = 0 (1n + 10n2) + -1n(4n + -3) * n = 0 Reorder the terms: 1n + 10n2 + -1n(-3 + 4n) * n = 0 Reorder the terms for easier multiplication: 1n + 10n2 + -1n * n(-3 + 4n) = 0 Multiply n * n 1n + 10n2 + -1n2(-3 + 4n) = 0 1n + 10n2 + (-3 * -1n2 + 4n * -1n2) = 0 1n + 10n2 + (3n2 + -4n3) = 0 Combine like terms: 10n2 + 3n2 = 13n2 1n + 13n2 + -4n3 = 0 Solving 1n + 13n2 + -4n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n'. n(1 + 13n + -4n2) = 0Subproblem 1
Set the factor 'n' equal to zero and attempt to solve: Simplifying n = 0 Solving n = 0 Move all terms containing n to the left, all other terms to the right. Simplifying n = 0Subproblem 2
Set the factor '(1 + 13n + -4n2)' equal to zero and attempt to solve: Simplifying 1 + 13n + -4n2 = 0 Solving 1 + 13n + -4n2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.25 + -3.25n + n2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -3.25n + 0.25 + n2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -3.25n + n2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -3.25n + n2 = 0 + 0.25 -3.25n + n2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -3.25n + n2 = 0.25 The n term is -3.25n. Take half its coefficient (-1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. -3.25n + 2.640625 + n2 = 0.25 + 2.640625 Reorder the terms: 2.640625 + -3.25n + n2 = 0.25 + 2.640625 Combine like terms: 0.25 + 2.640625 = 2.890625 2.640625 + -3.25n + n2 = 2.890625 Factor a perfect square on the left side: (n + -1.625)(n + -1.625) = 2.890625 Calculate the square root of the right side: 1.700183814 Break this problem into two subproblems by setting (n + -1.625) equal to 1.700183814 and -1.700183814.Subproblem 1
n + -1.625 = 1.700183814 Simplifying n + -1.625 = 1.700183814 Reorder the terms: -1.625 + n = 1.700183814 Solving -1.625 + n = 1.700183814 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + n = 1.700183814 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + n = 1.700183814 + 1.625 n = 1.700183814 + 1.625 Combine like terms: 1.700183814 + 1.625 = 3.325183814 n = 3.325183814 Simplifying n = 3.325183814Subproblem 2
n + -1.625 = -1.700183814 Simplifying n + -1.625 = -1.700183814 Reorder the terms: -1.625 + n = -1.700183814 Solving -1.625 + n = -1.700183814 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + n = -1.700183814 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + n = -1.700183814 + 1.625 n = -1.700183814 + 1.625 Combine like terms: -1.700183814 + 1.625 = -0.075183814 n = -0.075183814 Simplifying n = -0.075183814Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.325183814, -0.075183814}Solution
n = {0, 3.325183814, -0.075183814}
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