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Simplifying 4n2 + 6n + -103 = 0 Reorder the terms: -103 + 6n + 4n2 = 0 Solving -103 + 6n + 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'. -25.75 + 1.5n + n2 = 0 Move the constant term to the right: Add '25.75' to each side of the equation. -25.75 + 1.5n + 25.75 + n2 = 0 + 25.75 Reorder the terms: -25.75 + 25.75 + 1.5n + n2 = 0 + 25.75 Combine like terms: -25.75 + 25.75 = 0.00 0.00 + 1.5n + n2 = 0 + 25.75 1.5n + n2 = 0 + 25.75 Combine like terms: 0 + 25.75 = 25.75 1.5n + n2 = 25.75 The n term is 1.5n. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5n + 0.5625 + n2 = 25.75 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = 25.75 + 0.5625 Combine like terms: 25.75 + 0.5625 = 26.3125 0.5625 + 1.5n + n2 = 26.3125 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 26.3125 Calculate the square root of the right side: 5.129571132 Break this problem into two subproblems by setting (n + 0.75) equal to 5.129571132 and -5.129571132.Subproblem 1
n + 0.75 = 5.129571132 Simplifying n + 0.75 = 5.129571132 Reorder the terms: 0.75 + n = 5.129571132 Solving 0.75 + n = 5.129571132 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 5.129571132 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 5.129571132 + -0.75 n = 5.129571132 + -0.75 Combine like terms: 5.129571132 + -0.75 = 4.379571132 n = 4.379571132 Simplifying n = 4.379571132Subproblem 2
n + 0.75 = -5.129571132 Simplifying n + 0.75 = -5.129571132 Reorder the terms: 0.75 + n = -5.129571132 Solving 0.75 + n = -5.129571132 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -5.129571132 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -5.129571132 + -0.75 n = -5.129571132 + -0.75 Combine like terms: -5.129571132 + -0.75 = -5.879571132 n = -5.879571132 Simplifying n = -5.879571132Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.379571132, -5.879571132}
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