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Simplifying 10n2 + 20n + -8 = 0 Reorder the terms: -8 + 20n + 10n2 = 0 Solving -8 + 20n + 10n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + 10n + 5n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + 10n + 5n2)' equal to zero and attempt to solve: Simplifying -4 + 10n + 5n2 = 0 Solving -4 + 10n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.8 + 2n + n2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + 2n + 0.8 + n2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + 2n + n2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + 2n + n2 = 0 + 0.8 2n + n2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 2n + n2 = 0.8 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 0.8 + 1 Reorder the terms: 1 + 2n + n2 = 0.8 + 1 Combine like terms: 0.8 + 1 = 1.8 1 + 2n + n2 = 1.8 Factor a perfect square on the left side: (n + 1)(n + 1) = 1.8 Calculate the square root of the right side: 1.341640787 Break this problem into two subproblems by setting (n + 1) equal to 1.341640787 and -1.341640787.Subproblem 1
n + 1 = 1.341640787 Simplifying n + 1 = 1.341640787 Reorder the terms: 1 + n = 1.341640787 Solving 1 + n = 1.341640787 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 1.341640787 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.341640787 + -1 n = 1.341640787 + -1 Combine like terms: 1.341640787 + -1 = 0.341640787 n = 0.341640787 Simplifying n = 0.341640787Subproblem 2
n + 1 = -1.341640787 Simplifying n + 1 = -1.341640787 Reorder the terms: 1 + n = -1.341640787 Solving 1 + n = -1.341640787 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -1.341640787 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.341640787 + -1 n = -1.341640787 + -1 Combine like terms: -1.341640787 + -1 = -2.341640787 n = -2.341640787 Simplifying n = -2.341640787Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.341640787, -2.341640787}Solution
n = {0.341640787, -2.341640787}
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