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Simplifying -237 = -8(4n + 6) * 5n Reorder the terms: -237 = -8(6 + 4n) * 5n Reorder the terms for easier multiplication: -237 = -8 * 5n(6 + 4n) Multiply -8 * 5 -237 = -40n(6 + 4n) -237 = (6 * -40n + 4n * -40n) -237 = (-240n + -160n2) Solving -237 = -240n + -160n2 Solving for variable 'n'. Reorder the terms: -237 + 240n + 160n2 = -240n + 240n + -160n2 + 160n2 Combine like terms: -240n + 240n = 0 -237 + 240n + 160n2 = 0 + -160n2 + 160n2 -237 + 240n + 160n2 = -160n2 + 160n2 Combine like terms: -160n2 + 160n2 = 0 -237 + 240n + 160n2 = 0 Begin completing the square. Divide all terms by 160 the coefficient of the squared term: Divide each side by '160'. -1.48125 + 1.5n + n2 = 0 Move the constant term to the right: Add '1.48125' to each side of the equation. -1.48125 + 1.5n + 1.48125 + n2 = 0 + 1.48125 Reorder the terms: -1.48125 + 1.48125 + 1.5n + n2 = 0 + 1.48125 Combine like terms: -1.48125 + 1.48125 = 0.00000 0.00000 + 1.5n + n2 = 0 + 1.48125 1.5n + n2 = 0 + 1.48125 Combine like terms: 0 + 1.48125 = 1.48125 1.5n + n2 = 1.48125 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 = 1.48125 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = 1.48125 + 0.5625 Combine like terms: 1.48125 + 0.5625 = 2.04375 0.5625 + 1.5n + n2 = 2.04375 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 2.04375 Calculate the square root of the right side: 1.429597846 Break this problem into two subproblems by setting (n + 0.75) equal to 1.429597846 and -1.429597846.Subproblem 1
n + 0.75 = 1.429597846 Simplifying n + 0.75 = 1.429597846 Reorder the terms: 0.75 + n = 1.429597846 Solving 0.75 + n = 1.429597846 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 = 1.429597846 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 1.429597846 + -0.75 n = 1.429597846 + -0.75 Combine like terms: 1.429597846 + -0.75 = 0.679597846 n = 0.679597846 Simplifying n = 0.679597846Subproblem 2
n + 0.75 = -1.429597846 Simplifying n + 0.75 = -1.429597846 Reorder the terms: 0.75 + n = -1.429597846 Solving 0.75 + n = -1.429597846 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 = -1.429597846 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -1.429597846 + -0.75 n = -1.429597846 + -0.75 Combine like terms: -1.429597846 + -0.75 = -2.179597846 n = -2.179597846 Simplifying n = -2.179597846Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.679597846, -2.179597846}
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