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Simplifying 5n2 + 19n + -2 = 68 Reorder the terms: -2 + 19n + 5n2 = 68 Solving -2 + 19n + 5n2 = 68 Solving for variable 'n'. Reorder the terms: -2 + -68 + 19n + 5n2 = 68 + -68 Combine like terms: -2 + -68 = -70 -70 + 19n + 5n2 = 68 + -68 Combine like terms: 68 + -68 = 0 -70 + 19n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -14 + 3.8n + n2 = 0 Move the constant term to the right: Add '14' to each side of the equation. -14 + 3.8n + 14 + n2 = 0 + 14 Reorder the terms: -14 + 14 + 3.8n + n2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + 3.8n + n2 = 0 + 14 3.8n + n2 = 0 + 14 Combine like terms: 0 + 14 = 14 3.8n + n2 = 14 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 = 14 + 3.61 Reorder the terms: 3.61 + 3.8n + n2 = 14 + 3.61 Combine like terms: 14 + 3.61 = 17.61 3.61 + 3.8n + n2 = 17.61 Factor a perfect square on the left side: (n + 1.9)(n + 1.9) = 17.61 Calculate the square root of the right side: 4.196427052 Break this problem into two subproblems by setting (n + 1.9) equal to 4.196427052 and -4.196427052.Subproblem 1
n + 1.9 = 4.196427052 Simplifying n + 1.9 = 4.196427052 Reorder the terms: 1.9 + n = 4.196427052 Solving 1.9 + n = 4.196427052 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 = 4.196427052 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = 4.196427052 + -1.9 n = 4.196427052 + -1.9 Combine like terms: 4.196427052 + -1.9 = 2.296427052 n = 2.296427052 Simplifying n = 2.296427052Subproblem 2
n + 1.9 = -4.196427052 Simplifying n + 1.9 = -4.196427052 Reorder the terms: 1.9 + n = -4.196427052 Solving 1.9 + n = -4.196427052 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 = -4.196427052 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = -4.196427052 + -1.9 n = -4.196427052 + -1.9 Combine like terms: -4.196427052 + -1.9 = -6.096427052 n = -6.096427052 Simplifying n = -6.096427052Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.296427052, -6.096427052}
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