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Simplifying 2n2 + 19n + -68 = -2 Reorder the terms: -68 + 19n + 2n2 = -2 Solving -68 + 19n + 2n2 = -2 Solving for variable 'n'. Reorder the terms: -68 + 2 + 19n + 2n2 = -2 + 2 Combine like terms: -68 + 2 = -66 -66 + 19n + 2n2 = -2 + 2 Combine like terms: -2 + 2 = 0 -66 + 19n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -33 + 9.5n + n2 = 0 Move the constant term to the right: Add '33' to each side of the equation. -33 + 9.5n + 33 + n2 = 0 + 33 Reorder the terms: -33 + 33 + 9.5n + n2 = 0 + 33 Combine like terms: -33 + 33 = 0 0 + 9.5n + n2 = 0 + 33 9.5n + n2 = 0 + 33 Combine like terms: 0 + 33 = 33 9.5n + n2 = 33 The n term is 9.5n. Take half its coefficient (4.75). Square it (22.5625) and add it to both sides. Add '22.5625' to each side of the equation. 9.5n + 22.5625 + n2 = 33 + 22.5625 Reorder the terms: 22.5625 + 9.5n + n2 = 33 + 22.5625 Combine like terms: 33 + 22.5625 = 55.5625 22.5625 + 9.5n + n2 = 55.5625 Factor a perfect square on the left side: (n + 4.75)(n + 4.75) = 55.5625 Calculate the square root of the right side: 7.454025758 Break this problem into two subproblems by setting (n + 4.75) equal to 7.454025758 and -7.454025758.Subproblem 1
n + 4.75 = 7.454025758 Simplifying n + 4.75 = 7.454025758 Reorder the terms: 4.75 + n = 7.454025758 Solving 4.75 + n = 7.454025758 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + n = 7.454025758 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + n = 7.454025758 + -4.75 n = 7.454025758 + -4.75 Combine like terms: 7.454025758 + -4.75 = 2.704025758 n = 2.704025758 Simplifying n = 2.704025758Subproblem 2
n + 4.75 = -7.454025758 Simplifying n + 4.75 = -7.454025758 Reorder the terms: 4.75 + n = -7.454025758 Solving 4.75 + n = -7.454025758 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + n = -7.454025758 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + n = -7.454025758 + -4.75 n = -7.454025758 + -4.75 Combine like terms: -7.454025758 + -4.75 = -12.204025758 n = -12.204025758 Simplifying n = -12.204025758Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.704025758, -12.204025758}
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