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Simplifying 6n2 + -9n = 9 Reorder the terms: -9n + 6n2 = 9 Solving -9n + 6n2 = 9 Solving for variable 'n'. Reorder the terms: -9 + -9n + 6n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + -9n + 6n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-3 + -3n + 2n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-3 + -3n + 2n2)' equal to zero and attempt to solve: Simplifying -3 + -3n + 2n2 = 0 Solving -3 + -3n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + -1.5n + n2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -1.5n + 1.5 + n2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -1.5n + n2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -1.5n + n2 = 0 + 1.5 -1.5n + n2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -1.5n + n2 = 1.5 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = 1.5 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 1.5 + 0.5625 Combine like terms: 1.5 + 0.5625 = 2.0625 0.5625 + -1.5n + n2 = 2.0625 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (n + -0.75) equal to 1.436140662 and -1.436140662.Subproblem 1
n + -0.75 = 1.436140662 Simplifying n + -0.75 = 1.436140662 Reorder the terms: -0.75 + n = 1.436140662 Solving -0.75 + n = 1.436140662 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 1.436140662 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 1.436140662 + 0.75 n = 1.436140662 + 0.75 Combine like terms: 1.436140662 + 0.75 = 2.186140662 n = 2.186140662 Simplifying n = 2.186140662Subproblem 2
n + -0.75 = -1.436140662 Simplifying n + -0.75 = -1.436140662 Reorder the terms: -0.75 + n = -1.436140662 Solving -0.75 + n = -1.436140662 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -1.436140662 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -1.436140662 + 0.75 n = -1.436140662 + 0.75 Combine like terms: -1.436140662 + 0.75 = -0.686140662 n = -0.686140662 Simplifying n = -0.686140662Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.186140662, -0.686140662}Solution
n = {2.186140662, -0.686140662}
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