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Simplifying 2n2 + -2n + 5 = 400 Reorder the terms: 5 + -2n + 2n2 = 400 Solving 5 + -2n + 2n2 = 400 Solving for variable 'n'. Reorder the terms: 5 + -400 + -2n + 2n2 = 400 + -400 Combine like terms: 5 + -400 = -395 -395 + -2n + 2n2 = 400 + -400 Combine like terms: 400 + -400 = 0 -395 + -2n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -197.5 + -1n + n2 = 0 Move the constant term to the right: Add '197.5' to each side of the equation. -197.5 + -1n + 197.5 + n2 = 0 + 197.5 Reorder the terms: -197.5 + 197.5 + -1n + n2 = 0 + 197.5 Combine like terms: -197.5 + 197.5 = 0.0 0.0 + -1n + n2 = 0 + 197.5 -1n + n2 = 0 + 197.5 Combine like terms: 0 + 197.5 = 197.5 -1n + n2 = 197.5 The n term is -1n. 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. -1n + 0.25 + n2 = 197.5 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 197.5 + 0.25 Combine like terms: 197.5 + 0.25 = 197.75 0.25 + -1n + n2 = 197.75 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 197.75 Calculate the square root of the right side: 14.06236111 Break this problem into two subproblems by setting (n + -0.5) equal to 14.06236111 and -14.06236111.Subproblem 1
n + -0.5 = 14.06236111 Simplifying n + -0.5 = 14.06236111 Reorder the terms: -0.5 + n = 14.06236111 Solving -0.5 + n = 14.06236111 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 = 14.06236111 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 14.06236111 + 0.5 n = 14.06236111 + 0.5 Combine like terms: 14.06236111 + 0.5 = 14.56236111 n = 14.56236111 Simplifying n = 14.56236111Subproblem 2
n + -0.5 = -14.06236111 Simplifying n + -0.5 = -14.06236111 Reorder the terms: -0.5 + n = -14.06236111 Solving -0.5 + n = -14.06236111 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 = -14.06236111 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -14.06236111 + 0.5 n = -14.06236111 + 0.5 Combine like terms: -14.06236111 + 0.5 = -13.56236111 n = -13.56236111 Simplifying n = -13.56236111Solution
The solution to the problem is based on the solutions from the subproblems. n = {14.56236111, -13.56236111}
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