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Simplifying -3n(2n + 1) + 7 = -5 Reorder the terms: -3n(1 + 2n) + 7 = -5 (1 * -3n + 2n * -3n) + 7 = -5 (-3n + -6n2) + 7 = -5 Reorder the terms: 7 + -3n + -6n2 = -5 Solving 7 + -3n + -6n2 = -5 Solving for variable 'n'. Reorder the terms: 7 + 5 + -3n + -6n2 = -5 + 5 Combine like terms: 7 + 5 = 12 12 + -3n + -6n2 = -5 + 5 Combine like terms: -5 + 5 = 0 12 + -3n + -6n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(4 + -1n + -2n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(4 + -1n + -2n2)' equal to zero and attempt to solve: Simplifying 4 + -1n + -2n2 = 0 Solving 4 + -1n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2 + 0.5n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.5n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 0.5n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.5n + n2 = 0 + 2 0.5n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.5n + n2 = 2 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 = 2 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 2 + 0.0625 Combine like terms: 2 + 0.0625 = 2.0625 0.0625 + 0.5n + n2 = 2.0625 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (n + 0.25) equal to 1.436140662 and -1.436140662.Subproblem 1
n + 0.25 = 1.436140662 Simplifying n + 0.25 = 1.436140662 Reorder the terms: 0.25 + n = 1.436140662 Solving 0.25 + n = 1.436140662 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 = 1.436140662 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.436140662 + -0.25 n = 1.436140662 + -0.25 Combine like terms: 1.436140662 + -0.25 = 1.186140662 n = 1.186140662 Simplifying n = 1.186140662Subproblem 2
n + 0.25 = -1.436140662 Simplifying n + 0.25 = -1.436140662 Reorder the terms: 0.25 + n = -1.436140662 Solving 0.25 + n = -1.436140662 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 = -1.436140662 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.436140662 + -0.25 n = -1.436140662 + -0.25 Combine like terms: -1.436140662 + -0.25 = -1.686140662 n = -1.686140662 Simplifying n = -1.686140662Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.186140662, -1.686140662}Solution
n = {1.186140662, -1.686140662}
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