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Simplifying 3n2 + -12n = 19 Reorder the terms: -12n + 3n2 = 19 Solving -12n + 3n2 = 19 Solving for variable 'n'. Reorder the terms: -19 + -12n + 3n2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + -12n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6.333333333 + -4n + n2 = 0 Move the constant term to the right: Add '6.333333333' to each side of the equation. -6.333333333 + -4n + 6.333333333 + n2 = 0 + 6.333333333 Reorder the terms: -6.333333333 + 6.333333333 + -4n + n2 = 0 + 6.333333333 Combine like terms: -6.333333333 + 6.333333333 = 0.000000000 0.000000000 + -4n + n2 = 0 + 6.333333333 -4n + n2 = 0 + 6.333333333 Combine like terms: 0 + 6.333333333 = 6.333333333 -4n + n2 = 6.333333333 The n term is -4n. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4n + 4 + n2 = 6.333333333 + 4 Reorder the terms: 4 + -4n + n2 = 6.333333333 + 4 Combine like terms: 6.333333333 + 4 = 10.333333333 4 + -4n + n2 = 10.333333333 Factor a perfect square on the left side: (n + -2)(n + -2) = 10.333333333 Calculate the square root of the right side: 3.214550254 Break this problem into two subproblems by setting (n + -2) equal to 3.214550254 and -3.214550254.Subproblem 1
n + -2 = 3.214550254 Simplifying n + -2 = 3.214550254 Reorder the terms: -2 + n = 3.214550254 Solving -2 + n = 3.214550254 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 3.214550254 + 2 Combine like terms: -2 + 2 = 0 0 + n = 3.214550254 + 2 n = 3.214550254 + 2 Combine like terms: 3.214550254 + 2 = 5.214550254 n = 5.214550254 Simplifying n = 5.214550254Subproblem 2
n + -2 = -3.214550254 Simplifying n + -2 = -3.214550254 Reorder the terms: -2 + n = -3.214550254 Solving -2 + n = -3.214550254 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -3.214550254 + 2 Combine like terms: -2 + 2 = 0 0 + n = -3.214550254 + 2 n = -3.214550254 + 2 Combine like terms: -3.214550254 + 2 = -1.214550254 n = -1.214550254 Simplifying n = -1.214550254Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.214550254, -1.214550254}
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