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Simplifying 4n2 + -14 = -2n Reorder the terms: -14 + 4n2 = -2n Solving -14 + 4n2 = -2n Solving for variable 'n'. Reorder the terms: -14 + 2n + 4n2 = -2n + 2n Combine like terms: -2n + 2n = 0 -14 + 2n + 4n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + n + 2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + n + 2n2)' equal to zero and attempt to solve: Simplifying -7 + n + 2n2 = 0 Solving -7 + n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 0.5n + n2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 0.5n + 3.5 + n2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 0.5n + n2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 0.5n + n2 = 0 + 3.5 0.5n + n2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 0.5n + n2 = 3.5 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5n + 0.25 + n2 = 3.5 + 0.25 Reorder the terms: 0.25 + 0.5n + n2 = 3.5 + 0.25 Combine like terms: 3.5 + 0.25 = 3.75 0.25 + 0.5n + n2 = 3.75 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (n + 0.5) equal to 1.936491673 and -1.936491673.Subproblem 1
n + 0.5 = 1.936491673 Simplifying n + 0.5 = 1.936491673 Reorder the terms: 0.5 + n = 1.936491673 Solving 0.5 + n = 1.936491673 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.936491673 + -0.5 n = 1.936491673 + -0.5 Combine like terms: 1.936491673 + -0.5 = 1.436491673 n = 1.436491673 Simplifying n = 1.436491673Subproblem 2
n + 0.5 = -1.936491673 Simplifying n + 0.5 = -1.936491673 Reorder the terms: 0.5 + n = -1.936491673 Solving 0.5 + n = -1.936491673 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.936491673 + -0.5 n = -1.936491673 + -0.5 Combine like terms: -1.936491673 + -0.5 = -2.436491673 n = -2.436491673 Simplifying n = -2.436491673Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.436491673, -2.436491673}Solution
n = {1.436491673, -2.436491673}
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