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Simplifying 4(2n + -3) * n = 5 Reorder the terms: 4(-3 + 2n) * n = 5 Reorder the terms for easier multiplication: 4n(-3 + 2n) = 5 (-3 * 4n + 2n * 4n) = 5 (-12n + 8n2) = 5 Solving -12n + 8n2 = 5 Solving for variable 'n'. Reorder the terms: -5 + -12n + 8n2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + -12n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.625 + -1.5n + n2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + -1.5n + 0.625 + n2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + -1.5n + n2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + -1.5n + n2 = 0 + 0.625 -1.5n + n2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 -1.5n + n2 = 0.625 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 = 0.625 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 0.625 + 0.5625 Combine like terms: 0.625 + 0.5625 = 1.1875 0.5625 + -1.5n + n2 = 1.1875 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 1.1875 Calculate the square root of the right side: 1.089724736 Break this problem into two subproblems by setting (n + -0.75) equal to 1.089724736 and -1.089724736.Subproblem 1
n + -0.75 = 1.089724736 Simplifying n + -0.75 = 1.089724736 Reorder the terms: -0.75 + n = 1.089724736 Solving -0.75 + n = 1.089724736 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 = 1.089724736 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 1.089724736 + 0.75 n = 1.089724736 + 0.75 Combine like terms: 1.089724736 + 0.75 = 1.839724736 n = 1.839724736 Simplifying n = 1.839724736Subproblem 2
n + -0.75 = -1.089724736 Simplifying n + -0.75 = -1.089724736 Reorder the terms: -0.75 + n = -1.089724736 Solving -0.75 + n = -1.089724736 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 = -1.089724736 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -1.089724736 + 0.75 n = -1.089724736 + 0.75 Combine like terms: -1.089724736 + 0.75 = -0.339724736 n = -0.339724736 Simplifying n = -0.339724736Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.839724736, -0.339724736}
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