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Simplifying 3n2 + -21n + -128 = -5n + -4n2 Reorder the terms: -128 + -21n + 3n2 = -5n + -4n2 Solving -128 + -21n + 3n2 = -5n + -4n2 Solving for variable 'n'. Reorder the terms: -128 + -21n + 5n + 3n2 + 4n2 = -5n + -4n2 + 5n + 4n2 Combine like terms: -21n + 5n = -16n -128 + -16n + 3n2 + 4n2 = -5n + -4n2 + 5n + 4n2 Combine like terms: 3n2 + 4n2 = 7n2 -128 + -16n + 7n2 = -5n + -4n2 + 5n + 4n2 Reorder the terms: -128 + -16n + 7n2 = -5n + 5n + -4n2 + 4n2 Combine like terms: -5n + 5n = 0 -128 + -16n + 7n2 = 0 + -4n2 + 4n2 -128 + -16n + 7n2 = -4n2 + 4n2 Combine like terms: -4n2 + 4n2 = 0 -128 + -16n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -18.28571429 + -2.285714286n + n2 = 0 Move the constant term to the right: Add '18.28571429' to each side of the equation. -18.28571429 + -2.285714286n + 18.28571429 + n2 = 0 + 18.28571429 Reorder the terms: -18.28571429 + 18.28571429 + -2.285714286n + n2 = 0 + 18.28571429 Combine like terms: -18.28571429 + 18.28571429 = 0.00000000 0.00000000 + -2.285714286n + n2 = 0 + 18.28571429 -2.285714286n + n2 = 0 + 18.28571429 Combine like terms: 0 + 18.28571429 = 18.28571429 -2.285714286n + n2 = 18.28571429 The n term is -2.285714286n. Take half its coefficient (-1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. -2.285714286n + 1.306122449 + n2 = 18.28571429 + 1.306122449 Reorder the terms: 1.306122449 + -2.285714286n + n2 = 18.28571429 + 1.306122449 Combine like terms: 18.28571429 + 1.306122449 = 19.591836739 1.306122449 + -2.285714286n + n2 = 19.591836739 Factor a perfect square on the left side: (n + -1.142857143)(n + -1.142857143) = 19.591836739 Calculate the square root of the right side: 4.426266682 Break this problem into two subproblems by setting (n + -1.142857143) equal to 4.426266682 and -4.426266682.Subproblem 1
n + -1.142857143 = 4.426266682 Simplifying n + -1.142857143 = 4.426266682 Reorder the terms: -1.142857143 + n = 4.426266682 Solving -1.142857143 + n = 4.426266682 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + n = 4.426266682 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + n = 4.426266682 + 1.142857143 n = 4.426266682 + 1.142857143 Combine like terms: 4.426266682 + 1.142857143 = 5.569123825 n = 5.569123825 Simplifying n = 5.569123825Subproblem 2
n + -1.142857143 = -4.426266682 Simplifying n + -1.142857143 = -4.426266682 Reorder the terms: -1.142857143 + n = -4.426266682 Solving -1.142857143 + n = -4.426266682 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + n = -4.426266682 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + n = -4.426266682 + 1.142857143 n = -4.426266682 + 1.142857143 Combine like terms: -4.426266682 + 1.142857143 = -3.283409539 n = -3.283409539 Simplifying n = -3.283409539Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.569123825, -3.283409539}
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