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Simplifying 28n2 + 148n + 96 = 0 Reorder the terms: 96 + 148n + 28n2 = 0 Solving 96 + 148n + 28n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4'. 4(24 + 37n + 7n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(24 + 37n + 7n2)' equal to zero and attempt to solve: Simplifying 24 + 37n + 7n2 = 0 Solving 24 + 37n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 3.428571429 + 5.285714286n + n2 = 0 Move the constant term to the right: Add '-3.428571429' to each side of the equation. 3.428571429 + 5.285714286n + -3.428571429 + n2 = 0 + -3.428571429 Reorder the terms: 3.428571429 + -3.428571429 + 5.285714286n + n2 = 0 + -3.428571429 Combine like terms: 3.428571429 + -3.428571429 = 0.000000000 0.000000000 + 5.285714286n + n2 = 0 + -3.428571429 5.285714286n + n2 = 0 + -3.428571429 Combine like terms: 0 + -3.428571429 = -3.428571429 5.285714286n + n2 = -3.428571429 The n term is 5.285714286n. Take half its coefficient (2.642857143). Square it (6.984693878) and add it to both sides. Add '6.984693878' to each side of the equation. 5.285714286n + 6.984693878 + n2 = -3.428571429 + 6.984693878 Reorder the terms: 6.984693878 + 5.285714286n + n2 = -3.428571429 + 6.984693878 Combine like terms: -3.428571429 + 6.984693878 = 3.556122449 6.984693878 + 5.285714286n + n2 = 3.556122449 Factor a perfect square on the left side: (n + 2.642857143)(n + 2.642857143) = 3.556122449 Calculate the square root of the right side: 1.885768397 Break this problem into two subproblems by setting (n + 2.642857143) equal to 1.885768397 and -1.885768397.Subproblem 1
n + 2.642857143 = 1.885768397 Simplifying n + 2.642857143 = 1.885768397 Reorder the terms: 2.642857143 + n = 1.885768397 Solving 2.642857143 + n = 1.885768397 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.642857143' to each side of the equation. 2.642857143 + -2.642857143 + n = 1.885768397 + -2.642857143 Combine like terms: 2.642857143 + -2.642857143 = 0.000000000 0.000000000 + n = 1.885768397 + -2.642857143 n = 1.885768397 + -2.642857143 Combine like terms: 1.885768397 + -2.642857143 = -0.757088746 n = -0.757088746 Simplifying n = -0.757088746Subproblem 2
n + 2.642857143 = -1.885768397 Simplifying n + 2.642857143 = -1.885768397 Reorder the terms: 2.642857143 + n = -1.885768397 Solving 2.642857143 + n = -1.885768397 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.642857143' to each side of the equation. 2.642857143 + -2.642857143 + n = -1.885768397 + -2.642857143 Combine like terms: 2.642857143 + -2.642857143 = 0.000000000 0.000000000 + n = -1.885768397 + -2.642857143 n = -1.885768397 + -2.642857143 Combine like terms: -1.885768397 + -2.642857143 = -4.52862554 n = -4.52862554 Simplifying n = -4.52862554Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.757088746, -4.52862554}Solution
n = {-0.757088746, -4.52862554}
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