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Simplifying 13 = 2n2 + -3n Reorder the terms: 13 = -3n + 2n2 Solving 13 = -3n + 2n2 Solving for variable 'n'. Reorder the terms: 13 + 3n + -2n2 = -3n + 3n + 2n2 + -2n2 Combine like terms: -3n + 3n = 0 13 + 3n + -2n2 = 0 + 2n2 + -2n2 13 + 3n + -2n2 = 2n2 + -2n2 Combine like terms: 2n2 + -2n2 = 0 13 + 3n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -6.5 + -1.5n + n2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + -1.5n + 6.5 + n2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + -1.5n + n2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + -1.5n + n2 = 0 + 6.5 -1.5n + n2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 -1.5n + n2 = 6.5 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = 6.5 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 6.5 + 0.5625 Combine like terms: 6.5 + 0.5625 = 7.0625 0.5625 + -1.5n + n2 = 7.0625 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 7.0625 Calculate the square root of the right side: 2.657536453 Break this problem into two subproblems by setting (n + -0.75) equal to 2.657536453 and -2.657536453.Subproblem 1
n + -0.75 = 2.657536453 Simplifying n + -0.75 = 2.657536453 Reorder the terms: -0.75 + n = 2.657536453 Solving -0.75 + n = 2.657536453 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 2.657536453 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 2.657536453 + 0.75 n = 2.657536453 + 0.75 Combine like terms: 2.657536453 + 0.75 = 3.407536453 n = 3.407536453 Simplifying n = 3.407536453Subproblem 2
n + -0.75 = -2.657536453 Simplifying n + -0.75 = -2.657536453 Reorder the terms: -0.75 + n = -2.657536453 Solving -0.75 + n = -2.657536453 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -2.657536453 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -2.657536453 + 0.75 n = -2.657536453 + 0.75 Combine like terms: -2.657536453 + 0.75 = -1.907536453 n = -1.907536453 Simplifying n = -1.907536453Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.407536453, -1.907536453}
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