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Simplifying 6n2 + -12n + -22 = 0 Reorder the terms: -22 + -12n + 6n2 = 0 Solving -22 + -12n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-11 + -6n + 3n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-11 + -6n + 3n2)' equal to zero and attempt to solve: Simplifying -11 + -6n + 3n2 = 0 Solving -11 + -6n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3.666666667 + -2n + n2 = 0 Move the constant term to the right: Add '3.666666667' to each side of the equation. -3.666666667 + -2n + 3.666666667 + n2 = 0 + 3.666666667 Reorder the terms: -3.666666667 + 3.666666667 + -2n + n2 = 0 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + -2n + n2 = 0 + 3.666666667 -2n + n2 = 0 + 3.666666667 Combine like terms: 0 + 3.666666667 = 3.666666667 -2n + n2 = 3.666666667 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 = 3.666666667 + 1 Reorder the terms: 1 + -2n + n2 = 3.666666667 + 1 Combine like terms: 3.666666667 + 1 = 4.666666667 1 + -2n + n2 = 4.666666667 Factor a perfect square on the left side: (n + -1)(n + -1) = 4.666666667 Calculate the square root of the right side: 2.1602469 Break this problem into two subproblems by setting (n + -1) equal to 2.1602469 and -2.1602469.Subproblem 1
n + -1 = 2.1602469 Simplifying n + -1 = 2.1602469 Reorder the terms: -1 + n = 2.1602469 Solving -1 + n = 2.1602469 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 = 2.1602469 + 1 Combine like terms: -1 + 1 = 0 0 + n = 2.1602469 + 1 n = 2.1602469 + 1 Combine like terms: 2.1602469 + 1 = 3.1602469 n = 3.1602469 Simplifying n = 3.1602469Subproblem 2
n + -1 = -2.1602469 Simplifying n + -1 = -2.1602469 Reorder the terms: -1 + n = -2.1602469 Solving -1 + n = -2.1602469 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 = -2.1602469 + 1 Combine like terms: -1 + 1 = 0 0 + n = -2.1602469 + 1 n = -2.1602469 + 1 Combine like terms: -2.1602469 + 1 = -1.1602469 n = -1.1602469 Simplifying n = -1.1602469Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.1602469, -1.1602469}Solution
n = {3.1602469, -1.1602469}
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