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Simplifying 3n + -2n(n + 2) = -11 Reorder the terms: 3n + -2n(2 + n) = -11 3n + (2 * -2n + n * -2n) = -11 3n + (-4n + -2n2) = -11 Combine like terms: 3n + -4n = -1n -1n + -2n2 = -11 Solving -1n + -2n2 = -11 Solving for variable 'n'. Reorder the terms: 11 + -1n + -2n2 = -11 + 11 Combine like terms: -11 + 11 = 0 11 + -1n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -5.5 + 0.5n + n2 = 0 Move the constant term to the right: Add '5.5' to each side of the equation. -5.5 + 0.5n + 5.5 + n2 = 0 + 5.5 Reorder the terms: -5.5 + 5.5 + 0.5n + n2 = 0 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + 0.5n + n2 = 0 + 5.5 0.5n + n2 = 0 + 5.5 Combine like terms: 0 + 5.5 = 5.5 0.5n + n2 = 5.5 The n term is 0.5n. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5n + 0.0625 + n2 = 5.5 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 5.5 + 0.0625 Combine like terms: 5.5 + 0.0625 = 5.5625 0.0625 + 0.5n + n2 = 5.5625 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (n + 0.25) equal to 2.358495283 and -2.358495283.Subproblem 1
n + 0.25 = 2.358495283 Simplifying n + 0.25 = 2.358495283 Reorder the terms: 0.25 + n = 2.358495283 Solving 0.25 + n = 2.358495283 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 2.358495283 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 2.358495283 + -0.25 n = 2.358495283 + -0.25 Combine like terms: 2.358495283 + -0.25 = 2.108495283 n = 2.108495283 Simplifying n = 2.108495283Subproblem 2
n + 0.25 = -2.358495283 Simplifying n + 0.25 = -2.358495283 Reorder the terms: 0.25 + n = -2.358495283 Solving 0.25 + n = -2.358495283 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -2.358495283 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -2.358495283 + -0.25 n = -2.358495283 + -0.25 Combine like terms: -2.358495283 + -0.25 = -2.608495283 n = -2.608495283 Simplifying n = -2.608495283Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.108495283, -2.608495283}
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