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Simplifying -2n2 + -5n + 15 = 0 Reorder the terms: 15 + -5n + -2n2 = 0 Solving 15 + -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'. -7.5 + 2.5n + n2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 2.5n + 7.5 + n2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 2.5n + n2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 2.5n + n2 = 0 + 7.5 2.5n + n2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 2.5n + n2 = 7.5 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 = 7.5 + 1.5625 Reorder the terms: 1.5625 + 2.5n + n2 = 7.5 + 1.5625 Combine like terms: 7.5 + 1.5625 = 9.0625 1.5625 + 2.5n + n2 = 9.0625 Factor a perfect square on the left side: (n + 1.25)(n + 1.25) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (n + 1.25) equal to 3.010398645 and -3.010398645.Subproblem 1
n + 1.25 = 3.010398645 Simplifying n + 1.25 = 3.010398645 Reorder the terms: 1.25 + n = 3.010398645 Solving 1.25 + n = 3.010398645 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 = 3.010398645 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 3.010398645 + -1.25 n = 3.010398645 + -1.25 Combine like terms: 3.010398645 + -1.25 = 1.760398645 n = 1.760398645 Simplifying n = 1.760398645Subproblem 2
n + 1.25 = -3.010398645 Simplifying n + 1.25 = -3.010398645 Reorder the terms: 1.25 + n = -3.010398645 Solving 1.25 + n = -3.010398645 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 = -3.010398645 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -3.010398645 + -1.25 n = -3.010398645 + -1.25 Combine like terms: -3.010398645 + -1.25 = -4.260398645 n = -4.260398645 Simplifying n = -4.260398645Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.760398645, -4.260398645}
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