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Simplifying 4n2 + -7 = -7n Reorder the terms: -7 + 4n2 = -7n Solving -7 + 4n2 = -7n Solving for variable 'n'. Reorder the terms: -7 + 7n + 4n2 = -7n + 7n Combine like terms: -7n + 7n = 0 -7 + 7n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.75 + 1.75n + n2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + 1.75n + 1.75 + n2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + 1.75n + n2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + 1.75n + n2 = 0 + 1.75 1.75n + n2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 1.75n + n2 = 1.75 The n term is 1.75n. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75n + 0.765625 + n2 = 1.75 + 0.765625 Reorder the terms: 0.765625 + 1.75n + n2 = 1.75 + 0.765625 Combine like terms: 1.75 + 0.765625 = 2.515625 0.765625 + 1.75n + n2 = 2.515625 Factor a perfect square on the left side: (n + 0.875)(n + 0.875) = 2.515625 Calculate the square root of the right side: 1.586072193 Break this problem into two subproblems by setting (n + 0.875) equal to 1.586072193 and -1.586072193.Subproblem 1
n + 0.875 = 1.586072193 Simplifying n + 0.875 = 1.586072193 Reorder the terms: 0.875 + n = 1.586072193 Solving 0.875 + n = 1.586072193 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = 1.586072193 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = 1.586072193 + -0.875 n = 1.586072193 + -0.875 Combine like terms: 1.586072193 + -0.875 = 0.711072193 n = 0.711072193 Simplifying n = 0.711072193Subproblem 2
n + 0.875 = -1.586072193 Simplifying n + 0.875 = -1.586072193 Reorder the terms: 0.875 + n = -1.586072193 Solving 0.875 + n = -1.586072193 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = -1.586072193 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = -1.586072193 + -0.875 n = -1.586072193 + -0.875 Combine like terms: -1.586072193 + -0.875 = -2.461072193 n = -2.461072193 Simplifying n = -2.461072193Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.711072193, -2.461072193}
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