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Simplifying 5n2 + 19n + -68 = -28 Reorder the terms: -68 + 19n + 5n2 = -28 Solving -68 + 19n + 5n2 = -28 Solving for variable 'n'. Reorder the terms: -68 + 28 + 19n + 5n2 = -28 + 28 Combine like terms: -68 + 28 = -40 -40 + 19n + 5n2 = -28 + 28 Combine like terms: -28 + 28 = 0 -40 + 19n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -8 + 3.8n + n2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 3.8n + 8 + n2 = 0 + 8 Reorder the terms: -8 + 8 + 3.8n + n2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 3.8n + n2 = 0 + 8 3.8n + n2 = 0 + 8 Combine like terms: 0 + 8 = 8 3.8n + n2 = 8 The n term is 3.8n. Take half its coefficient (1.9). Square it (3.61) and add it to both sides. Add '3.61' to each side of the equation. 3.8n + 3.61 + n2 = 8 + 3.61 Reorder the terms: 3.61 + 3.8n + n2 = 8 + 3.61 Combine like terms: 8 + 3.61 = 11.61 3.61 + 3.8n + n2 = 11.61 Factor a perfect square on the left side: (n + 1.9)(n + 1.9) = 11.61 Calculate the square root of the right side: 3.407345007 Break this problem into two subproblems by setting (n + 1.9) equal to 3.407345007 and -3.407345007.Subproblem 1
n + 1.9 = 3.407345007 Simplifying n + 1.9 = 3.407345007 Reorder the terms: 1.9 + n = 3.407345007 Solving 1.9 + n = 3.407345007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + n = 3.407345007 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = 3.407345007 + -1.9 n = 3.407345007 + -1.9 Combine like terms: 3.407345007 + -1.9 = 1.507345007 n = 1.507345007 Simplifying n = 1.507345007Subproblem 2
n + 1.9 = -3.407345007 Simplifying n + 1.9 = -3.407345007 Reorder the terms: 1.9 + n = -3.407345007 Solving 1.9 + n = -3.407345007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + n = -3.407345007 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = -3.407345007 + -1.9 n = -3.407345007 + -1.9 Combine like terms: -3.407345007 + -1.9 = -5.307345007 n = -5.307345007 Simplifying n = -5.307345007Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.507345007, -5.307345007}
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