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Simplifying 0.3(n + -5) = 0.4 + -0.2n(n + -7) Reorder the terms: 0.3(-5 + n) = 0.4 + -0.2n(n + -7) (-5 * 0.3 + n * 0.3) = 0.4 + -0.2n(n + -7) (-1.5 + 0.3n) = 0.4 + -0.2n(n + -7) Reorder the terms: -1.5 + 0.3n = 0.4 + -0.2n(-7 + n) -1.5 + 0.3n = 0.4 + (-7 * -0.2n + n * -0.2n) -1.5 + 0.3n = 0.4 + (1.4n + -0.2n2) Solving -1.5 + 0.3n = 0.4 + 1.4n + -0.2n2 Solving for variable 'n'. Reorder the terms: -1.5 + -0.4 + 0.3n + -1.4n + 0.2n2 = 0.4 + 1.4n + -0.2n2 + -0.4 + -1.4n + 0.2n2 Combine like terms: -1.5 + -0.4 = -1.9 -1.9 + 0.3n + -1.4n + 0.2n2 = 0.4 + 1.4n + -0.2n2 + -0.4 + -1.4n + 0.2n2 Combine like terms: 0.3n + -1.4n = -1.1n -1.9 + -1.1n + 0.2n2 = 0.4 + 1.4n + -0.2n2 + -0.4 + -1.4n + 0.2n2 Reorder the terms: -1.9 + -1.1n + 0.2n2 = 0.4 + -0.4 + 1.4n + -1.4n + -0.2n2 + 0.2n2 Combine like terms: 0.4 + -0.4 = 0.0 -1.9 + -1.1n + 0.2n2 = 0.0 + 1.4n + -1.4n + -0.2n2 + 0.2n2 -1.9 + -1.1n + 0.2n2 = 1.4n + -1.4n + -0.2n2 + 0.2n2 Combine like terms: 1.4n + -1.4n = 0.0 -1.9 + -1.1n + 0.2n2 = 0.0 + -0.2n2 + 0.2n2 -1.9 + -1.1n + 0.2n2 = -0.2n2 + 0.2n2 Combine like terms: -0.2n2 + 0.2n2 = 0.0 -1.9 + -1.1n + 0.2n2 = 0.0 Begin completing the square. Divide all terms by 0.2 the coefficient of the squared term: Divide each side by '0.2'. -9.5 + -5.5n + n2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + -5.5n + 9.5 + n2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + -5.5n + n2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + -5.5n + n2 = 0 + 9.5 -5.5n + n2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 -5.5n + n2 = 9.5 The n term is -5.5n. Take half its coefficient (-2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. -5.5n + 7.5625 + n2 = 9.5 + 7.5625 Reorder the terms: 7.5625 + -5.5n + n2 = 9.5 + 7.5625 Combine like terms: 9.5 + 7.5625 = 17.0625 7.5625 + -5.5n + n2 = 17.0625 Factor a perfect square on the left side: (n + -2.75)(n + -2.75) = 17.0625 Calculate the square root of the right side: 4.13067791 Break this problem into two subproblems by setting (n + -2.75) equal to 4.13067791 and -4.13067791.Subproblem 1
n + -2.75 = 4.13067791 Simplifying n + -2.75 = 4.13067791 Reorder the terms: -2.75 + n = 4.13067791 Solving -2.75 + n = 4.13067791 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + n = 4.13067791 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + n = 4.13067791 + 2.75 n = 4.13067791 + 2.75 Combine like terms: 4.13067791 + 2.75 = 6.88067791 n = 6.88067791 Simplifying n = 6.88067791Subproblem 2
n + -2.75 = -4.13067791 Simplifying n + -2.75 = -4.13067791 Reorder the terms: -2.75 + n = -4.13067791 Solving -2.75 + n = -4.13067791 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + n = -4.13067791 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + n = -4.13067791 + 2.75 n = -4.13067791 + 2.75 Combine like terms: -4.13067791 + 2.75 = -1.38067791 n = -1.38067791 Simplifying n = -1.38067791Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.88067791, -1.38067791}
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