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Simplifying -2(n * n) + -5n + 16 = 0 Multiply n * n -2(n2) + -5n + 16 = 0 Reorder the terms: 16 + -5n + -2n2 = 0 Solving 16 + -5n + -2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -8 + 2.5n + n2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 2.5n + 8 + n2 = 0 + 8 Reorder the terms: -8 + 8 + 2.5n + n2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 2.5n + n2 = 0 + 8 2.5n + n2 = 0 + 8 Combine like terms: 0 + 8 = 8 2.5n + n2 = 8 The n term is 2.5n. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5n + 1.5625 + n2 = 8 + 1.5625 Reorder the terms: 1.5625 + 2.5n + n2 = 8 + 1.5625 Combine like terms: 8 + 1.5625 = 9.5625 1.5625 + 2.5n + n2 = 9.5625 Factor a perfect square on the left side: (n + 1.25)(n + 1.25) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (n + 1.25) equal to 3.092329219 and -3.092329219.Subproblem 1
n + 1.25 = 3.092329219 Simplifying n + 1.25 = 3.092329219 Reorder the terms: 1.25 + n = 3.092329219 Solving 1.25 + n = 3.092329219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = 3.092329219 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 3.092329219 + -1.25 n = 3.092329219 + -1.25 Combine like terms: 3.092329219 + -1.25 = 1.842329219 n = 1.842329219 Simplifying n = 1.842329219Subproblem 2
n + 1.25 = -3.092329219 Simplifying n + 1.25 = -3.092329219 Reorder the terms: 1.25 + n = -3.092329219 Solving 1.25 + n = -3.092329219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = -3.092329219 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -3.092329219 + -1.25 n = -3.092329219 + -1.25 Combine like terms: -3.092329219 + -1.25 = -4.342329219 n = -4.342329219 Simplifying n = -4.342329219Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.842329219, -4.342329219}
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