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Simplifying -12n2 = -9n + -8 Reorder the terms: -12n2 = -8 + -9n Solving -12n2 = -8 + -9n Solving for variable 'n'. Reorder the terms: 8 + 9n + -12n2 = -8 + -9n + 8 + 9n Reorder the terms: 8 + 9n + -12n2 = -8 + 8 + -9n + 9n Combine like terms: -8 + 8 = 0 8 + 9n + -12n2 = 0 + -9n + 9n 8 + 9n + -12n2 = -9n + 9n Combine like terms: -9n + 9n = 0 8 + 9n + -12n2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -0.6666666667 + -0.75n + n2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -0.75n + 0.6666666667 + n2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -0.75n + n2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -0.75n + n2 = 0 + 0.6666666667 -0.75n + n2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -0.75n + n2 = 0.6666666667 The n term is -0.75n. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75n + 0.140625 + n2 = 0.6666666667 + 0.140625 Reorder the terms: 0.140625 + -0.75n + n2 = 0.6666666667 + 0.140625 Combine like terms: 0.6666666667 + 0.140625 = 0.8072916667 0.140625 + -0.75n + n2 = 0.8072916667 Factor a perfect square on the left side: (n + -0.375)(n + -0.375) = 0.8072916667 Calculate the square root of the right side: 0.898494111 Break this problem into two subproblems by setting (n + -0.375) equal to 0.898494111 and -0.898494111.Subproblem 1
n + -0.375 = 0.898494111 Simplifying n + -0.375 = 0.898494111 Reorder the terms: -0.375 + n = 0.898494111 Solving -0.375 + n = 0.898494111 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = 0.898494111 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = 0.898494111 + 0.375 n = 0.898494111 + 0.375 Combine like terms: 0.898494111 + 0.375 = 1.273494111 n = 1.273494111 Simplifying n = 1.273494111Subproblem 2
n + -0.375 = -0.898494111 Simplifying n + -0.375 = -0.898494111 Reorder the terms: -0.375 + n = -0.898494111 Solving -0.375 + n = -0.898494111 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = -0.898494111 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = -0.898494111 + 0.375 n = -0.898494111 + 0.375 Combine like terms: -0.898494111 + 0.375 = -0.523494111 n = -0.523494111 Simplifying n = -0.523494111Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.273494111, -0.523494111}
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