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Simplifying 4n2 + -24n + 26 = 0 Reorder the terms: 26 + -24n + 4n2 = 0 Solving 26 + -24n + 4n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(13 + -12n + 2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(13 + -12n + 2n2)' equal to zero and attempt to solve: Simplifying 13 + -12n + 2n2 = 0 Solving 13 + -12n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 6.5 + -6n + n2 = 0 Move the constant term to the right: Add '-6.5' to each side of the equation. 6.5 + -6n + -6.5 + n2 = 0 + -6.5 Reorder the terms: 6.5 + -6.5 + -6n + n2 = 0 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + -6n + n2 = 0 + -6.5 -6n + n2 = 0 + -6.5 Combine like terms: 0 + -6.5 = -6.5 -6n + n2 = -6.5 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 = -6.5 + 9 Reorder the terms: 9 + -6n + n2 = -6.5 + 9 Combine like terms: -6.5 + 9 = 2.5 9 + -6n + n2 = 2.5 Factor a perfect square on the left side: (n + -3)(n + -3) = 2.5 Calculate the square root of the right side: 1.58113883 Break this problem into two subproblems by setting (n + -3) equal to 1.58113883 and -1.58113883.Subproblem 1
n + -3 = 1.58113883 Simplifying n + -3 = 1.58113883 Reorder the terms: -3 + n = 1.58113883 Solving -3 + n = 1.58113883 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 = 1.58113883 + 3 Combine like terms: -3 + 3 = 0 0 + n = 1.58113883 + 3 n = 1.58113883 + 3 Combine like terms: 1.58113883 + 3 = 4.58113883 n = 4.58113883 Simplifying n = 4.58113883Subproblem 2
n + -3 = -1.58113883 Simplifying n + -3 = -1.58113883 Reorder the terms: -3 + n = -1.58113883 Solving -3 + n = -1.58113883 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 = -1.58113883 + 3 Combine like terms: -3 + 3 = 0 0 + n = -1.58113883 + 3 n = -1.58113883 + 3 Combine like terms: -1.58113883 + 3 = 1.41886117 n = 1.41886117 Simplifying n = 1.41886117Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.58113883, 1.41886117}Solution
n = {4.58113883, 1.41886117}
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