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Simplifying 12n2 + 12n + -14 = 0 Reorder the terms: -14 + 12n + 12n2 = 0 Solving -14 + 12n + 12n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + 6n + 6n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + 6n + 6n2)' equal to zero and attempt to solve: Simplifying -7 + 6n + 6n2 = 0 Solving -7 + 6n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + n + n2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + n + 1.166666667 + n2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + n + n2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + n + n2 = 0 + 1.166666667 n + n2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 n + n2 = 1.166666667 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + n2 = 1.166666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 1.166666667 + 0.25 Combine like terms: 1.166666667 + 0.25 = 1.416666667 1.25 + n2 = 1.416666667 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 1.416666667 Calculate the square root of the right side: 1.190238072 Break this problem into two subproblems by setting (n + 0.5) equal to 1.190238072 and -1.190238072.Subproblem 1
n + 0.5 = 1.190238072 Simplifying n + 0.5 = 1.190238072 Reorder the terms: 0.5 + n = 1.190238072 Solving 0.5 + n = 1.190238072 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 1.190238072 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.190238072 + -0.5 n = 1.190238072 + -0.5 Combine like terms: 1.190238072 + -0.5 = 0.690238072 n = 0.690238072 Simplifying n = 0.690238072Subproblem 2
n + 0.5 = -1.190238072 Simplifying n + 0.5 = -1.190238072 Reorder the terms: 0.5 + n = -1.190238072 Solving 0.5 + n = -1.190238072 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -1.190238072 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.190238072 + -0.5 n = -1.190238072 + -0.5 Combine like terms: -1.190238072 + -0.5 = -1.690238072 n = -1.690238072 Simplifying n = -1.690238072Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.690238072, -1.690238072}Solution
n = {0.690238072, -1.690238072}
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