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Simplifying -5n(-8n + 5) + -5(3 + 4n) = 0 Reorder the terms: -5n(5 + -8n) + -5(3 + 4n) = 0 (5 * -5n + -8n * -5n) + -5(3 + 4n) = 0 (-25n + 40n2) + -5(3 + 4n) = 0 -25n + 40n2 + (3 * -5 + 4n * -5) = 0 -25n + 40n2 + (-15 + -20n) = 0 Reorder the terms: -15 + -25n + -20n + 40n2 = 0 Combine like terms: -25n + -20n = -45n -15 + -45n + 40n2 = 0 Solving -15 + -45n + 40n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '5'. 5(-3 + -9n + 8n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-3 + -9n + 8n2)' equal to zero and attempt to solve: Simplifying -3 + -9n + 8n2 = 0 Solving -3 + -9n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.375 + -1.125n + n2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -1.125n + 0.375 + n2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -1.125n + n2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -1.125n + n2 = 0 + 0.375 -1.125n + n2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -1.125n + n2 = 0.375 The n term is -1.125n. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125n + 0.31640625 + n2 = 0.375 + 0.31640625 Reorder the terms: 0.31640625 + -1.125n + n2 = 0.375 + 0.31640625 Combine like terms: 0.375 + 0.31640625 = 0.69140625 0.31640625 + -1.125n + n2 = 0.69140625 Factor a perfect square on the left side: (n + -0.5625)(n + -0.5625) = 0.69140625 Calculate the square root of the right side: 0.831508418 Break this problem into two subproblems by setting (n + -0.5625) equal to 0.831508418 and -0.831508418.Subproblem 1
n + -0.5625 = 0.831508418 Simplifying n + -0.5625 = 0.831508418 Reorder the terms: -0.5625 + n = 0.831508418 Solving -0.5625 + n = 0.831508418 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + n = 0.831508418 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = 0.831508418 + 0.5625 n = 0.831508418 + 0.5625 Combine like terms: 0.831508418 + 0.5625 = 1.394008418 n = 1.394008418 Simplifying n = 1.394008418Subproblem 2
n + -0.5625 = -0.831508418 Simplifying n + -0.5625 = -0.831508418 Reorder the terms: -0.5625 + n = -0.831508418 Solving -0.5625 + n = -0.831508418 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + n = -0.831508418 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = -0.831508418 + 0.5625 n = -0.831508418 + 0.5625 Combine like terms: -0.831508418 + 0.5625 = -0.269008418 n = -0.269008418 Simplifying n = -0.269008418Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.394008418, -0.269008418}Solution
n = {1.394008418, -0.269008418}
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