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Simplifying (2n2 + 3n) + -1(-6n2 + -7n + 4) = 0 Reorder the terms: (3n + 2n2) + -1(-6n2 + -7n + 4) = 0 Remove parenthesis around (3n + 2n2) 3n + 2n2 + -1(-6n2 + -7n + 4) = 0 Reorder the terms: 3n + 2n2 + -1(4 + -7n + -6n2) = 0 3n + 2n2 + (4 * -1 + -7n * -1 + -6n2 * -1) = 0 3n + 2n2 + (-4 + 7n + 6n2) = 0 Reorder the terms: -4 + 3n + 7n + 2n2 + 6n2 = 0 Combine like terms: 3n + 7n = 10n -4 + 10n + 2n2 + 6n2 = 0 Combine like terms: 2n2 + 6n2 = 8n2 -4 + 10n + 8n2 = 0 Solving -4 + 10n + 8n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-2 + 5n + 4n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-2 + 5n + 4n2)' equal to zero and attempt to solve: Simplifying -2 + 5n + 4n2 = 0 Solving -2 + 5n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + 1.25n + n2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 1.25n + 0.5 + n2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 1.25n + n2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 1.25n + n2 = 0 + 0.5 1.25n + n2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 1.25n + n2 = 0.5 The n term is 1.25n. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25n + 0.390625 + n2 = 0.5 + 0.390625 Reorder the terms: 0.390625 + 1.25n + n2 = 0.5 + 0.390625 Combine like terms: 0.5 + 0.390625 = 0.890625 0.390625 + 1.25n + n2 = 0.890625 Factor a perfect square on the left side: (n + 0.625)(n + 0.625) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (n + 0.625) equal to 0.943729304 and -0.943729304.Subproblem 1
n + 0.625 = 0.943729304 Simplifying n + 0.625 = 0.943729304 Reorder the terms: 0.625 + n = 0.943729304 Solving 0.625 + n = 0.943729304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + n = 0.943729304 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = 0.943729304 + -0.625 n = 0.943729304 + -0.625 Combine like terms: 0.943729304 + -0.625 = 0.318729304 n = 0.318729304 Simplifying n = 0.318729304Subproblem 2
n + 0.625 = -0.943729304 Simplifying n + 0.625 = -0.943729304 Reorder the terms: 0.625 + n = -0.943729304 Solving 0.625 + n = -0.943729304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + n = -0.943729304 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = -0.943729304 + -0.625 n = -0.943729304 + -0.625 Combine like terms: -0.943729304 + -0.625 = -1.568729304 n = -1.568729304 Simplifying n = -1.568729304Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.318729304, -1.568729304}Solution
n = {0.318729304, -1.568729304}
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