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Simplifying -9n + -10 = -3n(-5 + 4n) Reorder the terms: -10 + -9n = -3n(-5 + 4n) -10 + -9n = (-5 * -3n + 4n * -3n) -10 + -9n = (15n + -12n2) Solving -10 + -9n = 15n + -12n2 Solving for variable 'n'. Combine like terms: -9n + -15n = -24n -10 + -24n + 12n2 = 15n + -12n2 + -15n + 12n2 Reorder the terms: -10 + -24n + 12n2 = 15n + -15n + -12n2 + 12n2 Combine like terms: 15n + -15n = 0 -10 + -24n + 12n2 = 0 + -12n2 + 12n2 -10 + -24n + 12n2 = -12n2 + 12n2 Combine like terms: -12n2 + 12n2 = 0 -10 + -24n + 12n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -12n + 6n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -12n + 6n2)' equal to zero and attempt to solve: Simplifying -5 + -12n + 6n2 = 0 Solving -5 + -12n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.8333333333 + -2n + n2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + -2n + 0.8333333333 + n2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + -2n + n2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + -2n + n2 = 0 + 0.8333333333 -2n + n2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 -2n + n2 = 0.8333333333 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 0.8333333333 + 1 Reorder the terms: 1 + -2n + n2 = 0.8333333333 + 1 Combine like terms: 0.8333333333 + 1 = 1.8333333333 1 + -2n + n2 = 1.8333333333 Factor a perfect square on the left side: (n + -1)(n + -1) = 1.8333333333 Calculate the square root of the right side: 1.354006401 Break this problem into two subproblems by setting (n + -1) equal to 1.354006401 and -1.354006401.Subproblem 1
n + -1 = 1.354006401 Simplifying n + -1 = 1.354006401 Reorder the terms: -1 + n = 1.354006401 Solving -1 + n = 1.354006401 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.354006401 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.354006401 + 1 n = 1.354006401 + 1 Combine like terms: 1.354006401 + 1 = 2.354006401 n = 2.354006401 Simplifying n = 2.354006401Subproblem 2
n + -1 = -1.354006401 Simplifying n + -1 = -1.354006401 Reorder the terms: -1 + n = -1.354006401 Solving -1 + n = -1.354006401 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.354006401 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.354006401 + 1 n = -1.354006401 + 1 Combine like terms: -1.354006401 + 1 = -0.354006401 n = -0.354006401 Simplifying n = -0.354006401Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.354006401, -0.354006401}Solution
n = {2.354006401, -0.354006401}
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