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Simplifying 3n2 + 12n + -90 = -8 Reorder the terms: -90 + 12n + 3n2 = -8 Solving -90 + 12n + 3n2 = -8 Solving for variable 'n'. Reorder the terms: -90 + 8 + 12n + 3n2 = -8 + 8 Combine like terms: -90 + 8 = -82 -82 + 12n + 3n2 = -8 + 8 Combine like terms: -8 + 8 = 0 -82 + 12n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -27.33333333 + 4n + n2 = 0 Move the constant term to the right: Add '27.33333333' to each side of the equation. -27.33333333 + 4n + 27.33333333 + n2 = 0 + 27.33333333 Reorder the terms: -27.33333333 + 27.33333333 + 4n + n2 = 0 + 27.33333333 Combine like terms: -27.33333333 + 27.33333333 = 0.00000000 0.00000000 + 4n + n2 = 0 + 27.33333333 4n + n2 = 0 + 27.33333333 Combine like terms: 0 + 27.33333333 = 27.33333333 4n + n2 = 27.33333333 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = 27.33333333 + 4 Reorder the terms: 4 + 4n + n2 = 27.33333333 + 4 Combine like terms: 27.33333333 + 4 = 31.33333333 4 + 4n + n2 = 31.33333333 Factor a perfect square on the left side: (n + 2)(n + 2) = 31.33333333 Calculate the square root of the right side: 5.597618541 Break this problem into two subproblems by setting (n + 2) equal to 5.597618541 and -5.597618541.Subproblem 1
n + 2 = 5.597618541 Simplifying n + 2 = 5.597618541 Reorder the terms: 2 + n = 5.597618541 Solving 2 + n = 5.597618541 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 5.597618541 + -2 Combine like terms: 2 + -2 = 0 0 + n = 5.597618541 + -2 n = 5.597618541 + -2 Combine like terms: 5.597618541 + -2 = 3.597618541 n = 3.597618541 Simplifying n = 3.597618541Subproblem 2
n + 2 = -5.597618541 Simplifying n + 2 = -5.597618541 Reorder the terms: 2 + n = -5.597618541 Solving 2 + n = -5.597618541 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -5.597618541 + -2 Combine like terms: 2 + -2 = 0 0 + n = -5.597618541 + -2 n = -5.597618541 + -2 Combine like terms: -5.597618541 + -2 = -7.597618541 n = -7.597618541 Simplifying n = -7.597618541Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.597618541, -7.597618541}
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