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Simplifying 5n2 + 42n + 13 = 0 Reorder the terms: 13 + 42n + 5n2 = 0 Solving 13 + 42n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2.6 + 8.4n + n2 = 0 Move the constant term to the right: Add '-2.6' to each side of the equation. 2.6 + 8.4n + -2.6 + n2 = 0 + -2.6 Reorder the terms: 2.6 + -2.6 + 8.4n + n2 = 0 + -2.6 Combine like terms: 2.6 + -2.6 = 0.0 0.0 + 8.4n + n2 = 0 + -2.6 8.4n + n2 = 0 + -2.6 Combine like terms: 0 + -2.6 = -2.6 8.4n + n2 = -2.6 The n term is 8.4n. Take half its coefficient (4.2). Square it (17.64) and add it to both sides. Add '17.64' to each side of the equation. 8.4n + 17.64 + n2 = -2.6 + 17.64 Reorder the terms: 17.64 + 8.4n + n2 = -2.6 + 17.64 Combine like terms: -2.6 + 17.64 = 15.04 17.64 + 8.4n + n2 = 15.04 Factor a perfect square on the left side: (n + 4.2)(n + 4.2) = 15.04 Calculate the square root of the right side: 3.878143886 Break this problem into two subproblems by setting (n + 4.2) equal to 3.878143886 and -3.878143886.Subproblem 1
n + 4.2 = 3.878143886 Simplifying n + 4.2 = 3.878143886 Reorder the terms: 4.2 + n = 3.878143886 Solving 4.2 + n = 3.878143886 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.2' to each side of the equation. 4.2 + -4.2 + n = 3.878143886 + -4.2 Combine like terms: 4.2 + -4.2 = 0.0 0.0 + n = 3.878143886 + -4.2 n = 3.878143886 + -4.2 Combine like terms: 3.878143886 + -4.2 = -0.321856114 n = -0.321856114 Simplifying n = -0.321856114Subproblem 2
n + 4.2 = -3.878143886 Simplifying n + 4.2 = -3.878143886 Reorder the terms: 4.2 + n = -3.878143886 Solving 4.2 + n = -3.878143886 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.2' to each side of the equation. 4.2 + -4.2 + n = -3.878143886 + -4.2 Combine like terms: 4.2 + -4.2 = 0.0 0.0 + n = -3.878143886 + -4.2 n = -3.878143886 + -4.2 Combine like terms: -3.878143886 + -4.2 = -8.078143886 n = -8.078143886 Simplifying n = -8.078143886Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.321856114, -8.078143886}
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