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Simplifying 15n + -9 + -12n2 + 2 + -18 + 19n2 = 0 Reorder the terms: -9 + 2 + -18 + 15n + -12n2 + 19n2 = 0 Combine like terms: -9 + 2 = -7 -7 + -18 + 15n + -12n2 + 19n2 = 0 Combine like terms: -7 + -18 = -25 -25 + 15n + -12n2 + 19n2 = 0 Combine like terms: -12n2 + 19n2 = 7n2 -25 + 15n + 7n2 = 0 Solving -25 + 15n + 7n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -3.571428571 + 2.142857143n + n2 = 0 Move the constant term to the right: Add '3.571428571' to each side of the equation. -3.571428571 + 2.142857143n + 3.571428571 + n2 = 0 + 3.571428571 Reorder the terms: -3.571428571 + 3.571428571 + 2.142857143n + n2 = 0 + 3.571428571 Combine like terms: -3.571428571 + 3.571428571 = 0.000000000 0.000000000 + 2.142857143n + n2 = 0 + 3.571428571 2.142857143n + n2 = 0 + 3.571428571 Combine like terms: 0 + 3.571428571 = 3.571428571 2.142857143n + n2 = 3.571428571 The n term is 2.142857143n. Take half its coefficient (1.071428572). Square it (1.147959185) and add it to both sides. Add '1.147959185' to each side of the equation. 2.142857143n + 1.147959185 + n2 = 3.571428571 + 1.147959185 Reorder the terms: 1.147959185 + 2.142857143n + n2 = 3.571428571 + 1.147959185 Combine like terms: 3.571428571 + 1.147959185 = 4.719387756 1.147959185 + 2.142857143n + n2 = 4.719387756 Factor a perfect square on the left side: (n + 1.071428572)(n + 1.071428572) = 4.719387756 Calculate the square root of the right side: 2.17241519 Break this problem into two subproblems by setting (n + 1.071428572) equal to 2.17241519 and -2.17241519.Subproblem 1
n + 1.071428572 = 2.17241519 Simplifying n + 1.071428572 = 2.17241519 Reorder the terms: 1.071428572 + n = 2.17241519 Solving 1.071428572 + n = 2.17241519 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + n = 2.17241519 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + n = 2.17241519 + -1.071428572 n = 2.17241519 + -1.071428572 Combine like terms: 2.17241519 + -1.071428572 = 1.100986618 n = 1.100986618 Simplifying n = 1.100986618Subproblem 2
n + 1.071428572 = -2.17241519 Simplifying n + 1.071428572 = -2.17241519 Reorder the terms: 1.071428572 + n = -2.17241519 Solving 1.071428572 + n = -2.17241519 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + n = -2.17241519 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + n = -2.17241519 + -1.071428572 n = -2.17241519 + -1.071428572 Combine like terms: -2.17241519 + -1.071428572 = -3.243843762 n = -3.243843762 Simplifying n = -3.243843762Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.100986618, -3.243843762}
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