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Simplifying 12n2 = 3n + 5 Reorder the terms: 12n2 = 5 + 3n Solving 12n2 = 5 + 3n Solving for variable 'n'. Reorder the terms: -5 + -3n + 12n2 = 5 + 3n + -5 + -3n Reorder the terms: -5 + -3n + 12n2 = 5 + -5 + 3n + -3n Combine like terms: 5 + -5 = 0 -5 + -3n + 12n2 = 0 + 3n + -3n -5 + -3n + 12n2 = 3n + -3n Combine like terms: 3n + -3n = 0 -5 + -3n + 12n2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.4166666667 + -0.25n + n2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + -0.25n + 0.4166666667 + n2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + -0.25n + n2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + -0.25n + n2 = 0 + 0.4166666667 -0.25n + n2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 -0.25n + n2 = 0.4166666667 The n term is -0.25n. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25n + 0.015625 + n2 = 0.4166666667 + 0.015625 Reorder the terms: 0.015625 + -0.25n + n2 = 0.4166666667 + 0.015625 Combine like terms: 0.4166666667 + 0.015625 = 0.4322916667 0.015625 + -0.25n + n2 = 0.4322916667 Factor a perfect square on the left side: (n + -0.125)(n + -0.125) = 0.4322916667 Calculate the square root of the right side: 0.65748891 Break this problem into two subproblems by setting (n + -0.125) equal to 0.65748891 and -0.65748891.Subproblem 1
n + -0.125 = 0.65748891 Simplifying n + -0.125 = 0.65748891 Reorder the terms: -0.125 + n = 0.65748891 Solving -0.125 + n = 0.65748891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = 0.65748891 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = 0.65748891 + 0.125 n = 0.65748891 + 0.125 Combine like terms: 0.65748891 + 0.125 = 0.78248891 n = 0.78248891 Simplifying n = 0.78248891Subproblem 2
n + -0.125 = -0.65748891 Simplifying n + -0.125 = -0.65748891 Reorder the terms: -0.125 + n = -0.65748891 Solving -0.125 + n = -0.65748891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = -0.65748891 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = -0.65748891 + 0.125 n = -0.65748891 + 0.125 Combine like terms: -0.65748891 + 0.125 = -0.53248891 n = -0.53248891 Simplifying n = -0.53248891Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.78248891, -0.53248891}
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