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Simplifying 4n2 + 3n = 151 Reorder the terms: 3n + 4n2 = 151 Solving 3n + 4n2 = 151 Solving for variable 'n'. Reorder the terms: -151 + 3n + 4n2 = 151 + -151 Combine like terms: 151 + -151 = 0 -151 + 3n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -37.75 + 0.75n + n2 = 0 Move the constant term to the right: Add '37.75' to each side of the equation. -37.75 + 0.75n + 37.75 + n2 = 0 + 37.75 Reorder the terms: -37.75 + 37.75 + 0.75n + n2 = 0 + 37.75 Combine like terms: -37.75 + 37.75 = 0.00 0.00 + 0.75n + n2 = 0 + 37.75 0.75n + n2 = 0 + 37.75 Combine like terms: 0 + 37.75 = 37.75 0.75n + n2 = 37.75 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 = 37.75 + 0.140625 Reorder the terms: 0.140625 + 0.75n + n2 = 37.75 + 0.140625 Combine like terms: 37.75 + 0.140625 = 37.890625 0.140625 + 0.75n + n2 = 37.890625 Factor a perfect square on the left side: (n + 0.375)(n + 0.375) = 37.890625 Calculate the square root of the right side: 6.155536126 Break this problem into two subproblems by setting (n + 0.375) equal to 6.155536126 and -6.155536126.Subproblem 1
n + 0.375 = 6.155536126 Simplifying n + 0.375 = 6.155536126 Reorder the terms: 0.375 + n = 6.155536126 Solving 0.375 + n = 6.155536126 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 = 6.155536126 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = 6.155536126 + -0.375 n = 6.155536126 + -0.375 Combine like terms: 6.155536126 + -0.375 = 5.780536126 n = 5.780536126 Simplifying n = 5.780536126Subproblem 2
n + 0.375 = -6.155536126 Simplifying n + 0.375 = -6.155536126 Reorder the terms: 0.375 + n = -6.155536126 Solving 0.375 + n = -6.155536126 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 = -6.155536126 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = -6.155536126 + -0.375 n = -6.155536126 + -0.375 Combine like terms: -6.155536126 + -0.375 = -6.530536126 n = -6.530536126 Simplifying n = -6.530536126Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.780536126, -6.530536126}
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