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Simplifying 4n2 + -15n + 12 = 0 Reorder the terms: 12 + -15n + 4n2 = 0 Solving 12 + -15n + 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'. 3 + -3.75n + n2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -3.75n + -3 + n2 = 0 + -3 Reorder the terms: 3 + -3 + -3.75n + n2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -3.75n + n2 = 0 + -3 -3.75n + n2 = 0 + -3 Combine like terms: 0 + -3 = -3 -3.75n + n2 = -3 The n term is -3.75n. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75n + 3.515625 + n2 = -3 + 3.515625 Reorder the terms: 3.515625 + -3.75n + n2 = -3 + 3.515625 Combine like terms: -3 + 3.515625 = 0.515625 3.515625 + -3.75n + n2 = 0.515625 Factor a perfect square on the left side: (n + -1.875)(n + -1.875) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (n + -1.875) equal to 0.718070331 and -0.718070331.Subproblem 1
n + -1.875 = 0.718070331 Simplifying n + -1.875 = 0.718070331 Reorder the terms: -1.875 + n = 0.718070331 Solving -1.875 + n = 0.718070331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + n = 0.718070331 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + n = 0.718070331 + 1.875 n = 0.718070331 + 1.875 Combine like terms: 0.718070331 + 1.875 = 2.593070331 n = 2.593070331 Simplifying n = 2.593070331Subproblem 2
n + -1.875 = -0.718070331 Simplifying n + -1.875 = -0.718070331 Reorder the terms: -1.875 + n = -0.718070331 Solving -1.875 + n = -0.718070331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + n = -0.718070331 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + n = -0.718070331 + 1.875 n = -0.718070331 + 1.875 Combine like terms: -0.718070331 + 1.875 = 1.156929669 n = 1.156929669 Simplifying n = 1.156929669Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.593070331, 1.156929669}
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