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Simplifying 4n2 + -4n + -6 = 0 Reorder the terms: -6 + -4n + 4n2 = 0 Solving -6 + -4n + 4n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -2n + 2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -2n + 2n2)' equal to zero and attempt to solve: Simplifying -3 + -2n + 2n2 = 0 Solving -3 + -2n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + -1n + n2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -1n + 1.5 + n2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -1n + n2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -1n + n2 = 0 + 1.5 -1n + n2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -1n + n2 = 1.5 The n term is -1n. 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. -1n + 0.25 + n2 = 1.5 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 1.5 + 0.25 Combine like terms: 1.5 + 0.25 = 1.75 0.25 + -1n + n2 = 1.75 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (n + -0.5) equal to 1.322875656 and -1.322875656.Subproblem 1
n + -0.5 = 1.322875656 Simplifying n + -0.5 = 1.322875656 Reorder the terms: -0.5 + n = 1.322875656 Solving -0.5 + n = 1.322875656 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 = 1.322875656 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.322875656 + 0.5 n = 1.322875656 + 0.5 Combine like terms: 1.322875656 + 0.5 = 1.822875656 n = 1.822875656 Simplifying n = 1.822875656Subproblem 2
n + -0.5 = -1.322875656 Simplifying n + -0.5 = -1.322875656 Reorder the terms: -0.5 + n = -1.322875656 Solving -0.5 + n = -1.322875656 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 = -1.322875656 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.322875656 + 0.5 n = -1.322875656 + 0.5 Combine like terms: -1.322875656 + 0.5 = -0.822875656 n = -0.822875656 Simplifying n = -0.822875656Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.822875656, -0.822875656}Solution
n = {1.822875656, -0.822875656}
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