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Simplifying 4(n + 1) + -0.5n(6n + -4) = 0 Reorder the terms: 4(1 + n) + -0.5n(6n + -4) = 0 (1 * 4 + n * 4) + -0.5n(6n + -4) = 0 (4 + 4n) + -0.5n(6n + -4) = 0 Reorder the terms: 4 + 4n + -0.5n(-4 + 6n) = 0 4 + 4n + (-4 * -0.5n + 6n * -0.5n) = 0 4 + 4n + (2n + -3n2) = 0 Combine like terms: 4n + 2n = 6n 4 + 6n + -3n2 = 0 Solving 4 + 6n + -3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -1.333333333 + -2n + n2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -2n + 1.333333333 + n2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -2n + n2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -2n + n2 = 0 + 1.333333333 -2n + n2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -2n + n2 = 1.333333333 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 1.333333333 + 1 Reorder the terms: 1 + -2n + n2 = 1.333333333 + 1 Combine like terms: 1.333333333 + 1 = 2.333333333 1 + -2n + n2 = 2.333333333 Factor a perfect square on the left side: (n + -1)(n + -1) = 2.333333333 Calculate the square root of the right side: 1.527525232 Break this problem into two subproblems by setting (n + -1) equal to 1.527525232 and -1.527525232.Subproblem 1
n + -1 = 1.527525232 Simplifying n + -1 = 1.527525232 Reorder the terms: -1 + n = 1.527525232 Solving -1 + n = 1.527525232 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.527525232 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.527525232 + 1 n = 1.527525232 + 1 Combine like terms: 1.527525232 + 1 = 2.527525232 n = 2.527525232 Simplifying n = 2.527525232Subproblem 2
n + -1 = -1.527525232 Simplifying n + -1 = -1.527525232 Reorder the terms: -1 + n = -1.527525232 Solving -1 + n = -1.527525232 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.527525232 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.527525232 + 1 n = -1.527525232 + 1 Combine like terms: -1.527525232 + 1 = -0.527525232 n = -0.527525232 Simplifying n = -0.527525232Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.527525232, -0.527525232}
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