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Simplifying 10n2 + 6n = 12 Reorder the terms: 6n + 10n2 = 12 Solving 6n + 10n2 = 12 Solving for variable 'n'. Reorder the terms: -12 + 6n + 10n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + 6n + 10n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-6 + 3n + 5n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-6 + 3n + 5n2)' equal to zero and attempt to solve: Simplifying -6 + 3n + 5n2 = 0 Solving -6 + 3n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.2 + 0.6n + n2 = 0 Move the constant term to the right: Add '1.2' to each side of the equation. -1.2 + 0.6n + 1.2 + n2 = 0 + 1.2 Reorder the terms: -1.2 + 1.2 + 0.6n + n2 = 0 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + 0.6n + n2 = 0 + 1.2 0.6n + n2 = 0 + 1.2 Combine like terms: 0 + 1.2 = 1.2 0.6n + n2 = 1.2 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 = 1.2 + 0.09 Reorder the terms: 0.09 + 0.6n + n2 = 1.2 + 0.09 Combine like terms: 1.2 + 0.09 = 1.29 0.09 + 0.6n + n2 = 1.29 Factor a perfect square on the left side: (n + 0.3)(n + 0.3) = 1.29 Calculate the square root of the right side: 1.135781669 Break this problem into two subproblems by setting (n + 0.3) equal to 1.135781669 and -1.135781669.Subproblem 1
n + 0.3 = 1.135781669 Simplifying n + 0.3 = 1.135781669 Reorder the terms: 0.3 + n = 1.135781669 Solving 0.3 + n = 1.135781669 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 = 1.135781669 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = 1.135781669 + -0.3 n = 1.135781669 + -0.3 Combine like terms: 1.135781669 + -0.3 = 0.835781669 n = 0.835781669 Simplifying n = 0.835781669Subproblem 2
n + 0.3 = -1.135781669 Simplifying n + 0.3 = -1.135781669 Reorder the terms: 0.3 + n = -1.135781669 Solving 0.3 + n = -1.135781669 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 = -1.135781669 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = -1.135781669 + -0.3 n = -1.135781669 + -0.3 Combine like terms: -1.135781669 + -0.3 = -1.435781669 n = -1.435781669 Simplifying n = -1.435781669Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.835781669, -1.435781669}Solution
n = {0.835781669, -1.435781669}
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