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Simplifying 12n2 + -3n = 21 Reorder the terms: -3n + 12n2 = 21 Solving -3n + 12n2 = 21 Solving for variable 'n'. Reorder the terms: -21 + -3n + 12n2 = 21 + -21 Combine like terms: 21 + -21 = 0 -21 + -3n + 12n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-7 + -1n + 4n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-7 + -1n + 4n2)' equal to zero and attempt to solve: Simplifying -7 + -1n + 4n2 = 0 Solving -7 + -1n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.75 + -0.25n + n2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + -0.25n + 1.75 + n2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + -0.25n + n2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + -0.25n + n2 = 0 + 1.75 -0.25n + n2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 -0.25n + n2 = 1.75 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 = 1.75 + 0.015625 Reorder the terms: 0.015625 + -0.25n + n2 = 1.75 + 0.015625 Combine like terms: 1.75 + 0.015625 = 1.765625 0.015625 + -0.25n + n2 = 1.765625 Factor a perfect square on the left side: (n + -0.125)(n + -0.125) = 1.765625 Calculate the square root of the right side: 1.328768227 Break this problem into two subproblems by setting (n + -0.125) equal to 1.328768227 and -1.328768227.Subproblem 1
n + -0.125 = 1.328768227 Simplifying n + -0.125 = 1.328768227 Reorder the terms: -0.125 + n = 1.328768227 Solving -0.125 + n = 1.328768227 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 = 1.328768227 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = 1.328768227 + 0.125 n = 1.328768227 + 0.125 Combine like terms: 1.328768227 + 0.125 = 1.453768227 n = 1.453768227 Simplifying n = 1.453768227Subproblem 2
n + -0.125 = -1.328768227 Simplifying n + -0.125 = -1.328768227 Reorder the terms: -0.125 + n = -1.328768227 Solving -0.125 + n = -1.328768227 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 = -1.328768227 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = -1.328768227 + 0.125 n = -1.328768227 + 0.125 Combine like terms: -1.328768227 + 0.125 = -1.203768227 n = -1.203768227 Simplifying n = -1.203768227Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.453768227, -1.203768227}Solution
n = {1.453768227, -1.203768227}
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