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Simplifying 10n2 = -6n + 6 Reorder the terms: 10n2 = 6 + -6n Solving 10n2 = 6 + -6n Solving for variable 'n'. Reorder the terms: -6 + 6n + 10n2 = 6 + -6n + -6 + 6n Reorder the terms: -6 + 6n + 10n2 = 6 + -6 + -6n + 6n Combine like terms: 6 + -6 = 0 -6 + 6n + 10n2 = 0 + -6n + 6n -6 + 6n + 10n2 = -6n + 6n Combine like terms: -6n + 6n = 0 -6 + 6n + 10n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 3n + 5n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 3n + 5n2)' equal to zero and attempt to solve: Simplifying -3 + 3n + 5n2 = 0 Solving -3 + 3n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.6 + 0.6n + n2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + 0.6n + 0.6 + n2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + 0.6n + n2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + 0.6n + n2 = 0 + 0.6 0.6n + n2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 0.6n + n2 = 0.6 The n term is 0.6n. Take half its coefficient (0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. 0.6n + 0.09 + n2 = 0.6 + 0.09 Reorder the terms: 0.09 + 0.6n + n2 = 0.6 + 0.09 Combine like terms: 0.6 + 0.09 = 0.69 0.09 + 0.6n + n2 = 0.69 Factor a perfect square on the left side: (n + 0.3)(n + 0.3) = 0.69 Calculate the square root of the right side: 0.830662386 Break this problem into two subproblems by setting (n + 0.3) equal to 0.830662386 and -0.830662386.Subproblem 1
n + 0.3 = 0.830662386 Simplifying n + 0.3 = 0.830662386 Reorder the terms: 0.3 + n = 0.830662386 Solving 0.3 + n = 0.830662386 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + n = 0.830662386 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = 0.830662386 + -0.3 n = 0.830662386 + -0.3 Combine like terms: 0.830662386 + -0.3 = 0.530662386 n = 0.530662386 Simplifying n = 0.530662386Subproblem 2
n + 0.3 = -0.830662386 Simplifying n + 0.3 = -0.830662386 Reorder the terms: 0.3 + n = -0.830662386 Solving 0.3 + n = -0.830662386 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + n = -0.830662386 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = -0.830662386 + -0.3 n = -0.830662386 + -0.3 Combine like terms: -0.830662386 + -0.3 = -1.130662386 n = -1.130662386 Simplifying n = -1.130662386Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.530662386, -1.130662386}Solution
n = {0.530662386, -1.130662386}
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