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Simplifying n2 + -12n + -78 = -9 Reorder the terms: -78 + -12n + n2 = -9 Solving -78 + -12n + n2 = -9 Solving for variable 'n'. Reorder the terms: -78 + 9 + -12n + n2 = -9 + 9 Combine like terms: -78 + 9 = -69 -69 + -12n + n2 = -9 + 9 Combine like terms: -9 + 9 = 0 -69 + -12n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '69' to each side of the equation. -69 + -12n + 69 + n2 = 0 + 69 Reorder the terms: -69 + 69 + -12n + n2 = 0 + 69 Combine like terms: -69 + 69 = 0 0 + -12n + n2 = 0 + 69 -12n + n2 = 0 + 69 Combine like terms: 0 + 69 = 69 -12n + n2 = 69 The n term is -12n. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12n + 36 + n2 = 69 + 36 Reorder the terms: 36 + -12n + n2 = 69 + 36 Combine like terms: 69 + 36 = 105 36 + -12n + n2 = 105 Factor a perfect square on the left side: (n + -6)(n + -6) = 105 Calculate the square root of the right side: 10.246950766 Break this problem into two subproblems by setting (n + -6) equal to 10.246950766 and -10.246950766.Subproblem 1
n + -6 = 10.246950766 Simplifying n + -6 = 10.246950766 Reorder the terms: -6 + n = 10.246950766 Solving -6 + n = 10.246950766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = 10.246950766 + 6 Combine like terms: -6 + 6 = 0 0 + n = 10.246950766 + 6 n = 10.246950766 + 6 Combine like terms: 10.246950766 + 6 = 16.246950766 n = 16.246950766 Simplifying n = 16.246950766Subproblem 2
n + -6 = -10.246950766 Simplifying n + -6 = -10.246950766 Reorder the terms: -6 + n = -10.246950766 Solving -6 + n = -10.246950766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = -10.246950766 + 6 Combine like terms: -6 + 6 = 0 0 + n = -10.246950766 + 6 n = -10.246950766 + 6 Combine like terms: -10.246950766 + 6 = -4.246950766 n = -4.246950766 Simplifying n = -4.246950766Solution
The solution to the problem is based on the solutions from the subproblems. n = {16.246950766, -4.246950766}
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