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Simplifying 5n2 + 10n + -24 = -12 Reorder the terms: -24 + 10n + 5n2 = -12 Solving -24 + 10n + 5n2 = -12 Solving for variable 'n'. Reorder the terms: -24 + 12 + 10n + 5n2 = -12 + 12 Combine like terms: -24 + 12 = -12 -12 + 10n + 5n2 = -12 + 12 Combine like terms: -12 + 12 = 0 -12 + 10n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.4 + 2n + n2 = 0 Move the constant term to the right: Add '2.4' to each side of the equation. -2.4 + 2n + 2.4 + n2 = 0 + 2.4 Reorder the terms: -2.4 + 2.4 + 2n + n2 = 0 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + 2n + n2 = 0 + 2.4 2n + n2 = 0 + 2.4 Combine like terms: 0 + 2.4 = 2.4 2n + n2 = 2.4 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 2.4 + 1 Reorder the terms: 1 + 2n + n2 = 2.4 + 1 Combine like terms: 2.4 + 1 = 3.4 1 + 2n + n2 = 3.4 Factor a perfect square on the left side: (n + 1)(n + 1) = 3.4 Calculate the square root of the right side: 1.843908891 Break this problem into two subproblems by setting (n + 1) equal to 1.843908891 and -1.843908891.Subproblem 1
n + 1 = 1.843908891 Simplifying n + 1 = 1.843908891 Reorder the terms: 1 + n = 1.843908891 Solving 1 + n = 1.843908891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 1.843908891 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.843908891 + -1 n = 1.843908891 + -1 Combine like terms: 1.843908891 + -1 = 0.843908891 n = 0.843908891 Simplifying n = 0.843908891Subproblem 2
n + 1 = -1.843908891 Simplifying n + 1 = -1.843908891 Reorder the terms: 1 + n = -1.843908891 Solving 1 + n = -1.843908891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -1.843908891 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.843908891 + -1 n = -1.843908891 + -1 Combine like terms: -1.843908891 + -1 = -2.843908891 n = -2.843908891 Simplifying n = -2.843908891Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.843908891, -2.843908891}
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