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Simplifying 4n2 + 7n + 1 = 0 Reorder the terms: 1 + 7n + 4n2 = 0 Solving 1 + 7n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + 1.75n + n2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + 1.75n + -0.25 + n2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + 1.75n + n2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + 1.75n + n2 = 0 + -0.25 1.75n + n2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 1.75n + n2 = -0.25 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 = -0.25 + 0.765625 Reorder the terms: 0.765625 + 1.75n + n2 = -0.25 + 0.765625 Combine like terms: -0.25 + 0.765625 = 0.515625 0.765625 + 1.75n + n2 = 0.515625 Factor a perfect square on the left side: (n + 0.875)(n + 0.875) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (n + 0.875) equal to 0.718070331 and -0.718070331.Subproblem 1
n + 0.875 = 0.718070331 Simplifying n + 0.875 = 0.718070331 Reorder the terms: 0.875 + n = 0.718070331 Solving 0.875 + n = 0.718070331 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 = 0.718070331 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = 0.718070331 + -0.875 n = 0.718070331 + -0.875 Combine like terms: 0.718070331 + -0.875 = -0.156929669 n = -0.156929669 Simplifying n = -0.156929669Subproblem 2
n + 0.875 = -0.718070331 Simplifying n + 0.875 = -0.718070331 Reorder the terms: 0.875 + n = -0.718070331 Solving 0.875 + n = -0.718070331 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 = -0.718070331 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = -0.718070331 + -0.875 n = -0.718070331 + -0.875 Combine like terms: -0.718070331 + -0.875 = -1.593070331 n = -1.593070331 Simplifying n = -1.593070331Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.156929669, -1.593070331}
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