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Simplifying 3n2 + 13n + -3 = 0 Reorder the terms: -3 + 13n + 3n2 = 0 Solving -3 + 13n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + 4.333333333n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 4.333333333n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + 4.333333333n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 4.333333333n + n2 = 0 + 1 4.333333333n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 4.333333333n + n2 = 1 The n term is 4.333333333n. Take half its coefficient (2.166666667). Square it (4.694444446) and add it to both sides. Add '4.694444446' to each side of the equation. 4.333333333n + 4.694444446 + n2 = 1 + 4.694444446 Reorder the terms: 4.694444446 + 4.333333333n + n2 = 1 + 4.694444446 Combine like terms: 1 + 4.694444446 = 5.694444446 4.694444446 + 4.333333333n + n2 = 5.694444446 Factor a perfect square on the left side: (n + 2.166666667)(n + 2.166666667) = 5.694444446 Calculate the square root of the right side: 2.386303511 Break this problem into two subproblems by setting (n + 2.166666667) equal to 2.386303511 and -2.386303511.Subproblem 1
n + 2.166666667 = 2.386303511 Simplifying n + 2.166666667 = 2.386303511 Reorder the terms: 2.166666667 + n = 2.386303511 Solving 2.166666667 + n = 2.386303511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + n = 2.386303511 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + n = 2.386303511 + -2.166666667 n = 2.386303511 + -2.166666667 Combine like terms: 2.386303511 + -2.166666667 = 0.219636844 n = 0.219636844 Simplifying n = 0.219636844Subproblem 2
n + 2.166666667 = -2.386303511 Simplifying n + 2.166666667 = -2.386303511 Reorder the terms: 2.166666667 + n = -2.386303511 Solving 2.166666667 + n = -2.386303511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + n = -2.386303511 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + n = -2.386303511 + -2.166666667 n = -2.386303511 + -2.166666667 Combine like terms: -2.386303511 + -2.166666667 = -4.552970178 n = -4.552970178 Simplifying n = -4.552970178Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.219636844, -4.552970178}
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