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Simplifying 4n(n2 + -6n + 7) = 0 Reorder the terms: 4n(7 + -6n + n2) = 0 (7 * 4n + -6n * 4n + n2 * 4n) = 0 (28n + -24n2 + 4n3) = 0 Solving 28n + -24n2 + 4n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4n'. 4n(7 + -6n + n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor 'n' equal to zero and attempt to solve: Simplifying n = 0 Solving n = 0 Move all terms containing n to the left, all other terms to the right. Simplifying n = 0Subproblem 2
Set the factor '(7 + -6n + n2)' equal to zero and attempt to solve: Simplifying 7 + -6n + n2 = 0 Solving 7 + -6n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-7' to each side of the equation. 7 + -6n + -7 + n2 = 0 + -7 Reorder the terms: 7 + -7 + -6n + n2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -6n + n2 = 0 + -7 -6n + n2 = 0 + -7 Combine like terms: 0 + -7 = -7 -6n + n2 = -7 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = -7 + 9 Reorder the terms: 9 + -6n + n2 = -7 + 9 Combine like terms: -7 + 9 = 2 9 + -6n + n2 = 2 Factor a perfect square on the left side: (n + -3)(n + -3) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (n + -3) equal to 1.414213562 and -1.414213562.Subproblem 1
n + -3 = 1.414213562 Simplifying n + -3 = 1.414213562 Reorder the terms: -3 + n = 1.414213562 Solving -3 + n = 1.414213562 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + n = 1.414213562 + 3 n = 1.414213562 + 3 Combine like terms: 1.414213562 + 3 = 4.414213562 n = 4.414213562 Simplifying n = 4.414213562Subproblem 2
n + -3 = -1.414213562 Simplifying n + -3 = -1.414213562 Reorder the terms: -3 + n = -1.414213562 Solving -3 + n = -1.414213562 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + n = -1.414213562 + 3 n = -1.414213562 + 3 Combine like terms: -1.414213562 + 3 = 1.585786438 n = 1.585786438 Simplifying n = 1.585786438Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.414213562, 1.585786438}Solution
n = {0, 4.414213562, 1.585786438}
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