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Simplifying 2n2 + 2n = 19 Reorder the terms: 2n + 2n2 = 19 Solving 2n + 2n2 = 19 Solving for variable 'n'. Reorder the terms: -19 + 2n + 2n2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 2n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9.5 + n + n2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + n + 9.5 + n2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + n + n2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + n + n2 = 0 + 9.5 n + n2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 n + n2 = 9.5 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + n2 = 9.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 9.5 + 0.25 Combine like terms: 9.5 + 0.25 = 9.75 1.25 + n2 = 9.75 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (n + 0.5) equal to 3.122498999 and -3.122498999.Subproblem 1
n + 0.5 = 3.122498999 Simplifying n + 0.5 = 3.122498999 Reorder the terms: 0.5 + n = 3.122498999 Solving 0.5 + n = 3.122498999 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 3.122498999 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 3.122498999 + -0.5 n = 3.122498999 + -0.5 Combine like terms: 3.122498999 + -0.5 = 2.622498999 n = 2.622498999 Simplifying n = 2.622498999Subproblem 2
n + 0.5 = -3.122498999 Simplifying n + 0.5 = -3.122498999 Reorder the terms: 0.5 + n = -3.122498999 Solving 0.5 + n = -3.122498999 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -3.122498999 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -3.122498999 + -0.5 n = -3.122498999 + -0.5 Combine like terms: -3.122498999 + -0.5 = -3.622498999 n = -3.622498999 Simplifying n = -3.622498999Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.622498999, -3.622498999}
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