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Simplifying -3n + -1(8n + -7) = n(n + 4) Reorder the terms: -3n + -1(-7 + 8n) = n(n + 4) -3n + (-7 * -1 + 8n * -1) = n(n + 4) -3n + (7 + -8n) = n(n + 4) Reorder the terms: 7 + -3n + -8n = n(n + 4) Combine like terms: -3n + -8n = -11n 7 + -11n = n(n + 4) Reorder the terms: 7 + -11n = n(4 + n) 7 + -11n = (4 * n + n * n) 7 + -11n = (4n + n2) Solving 7 + -11n = 4n + n2 Solving for variable 'n'. Combine like terms: -11n + -4n = -15n 7 + -15n + -1n2 = 4n + n2 + -4n + -1n2 Reorder the terms: 7 + -15n + -1n2 = 4n + -4n + n2 + -1n2 Combine like terms: 4n + -4n = 0 7 + -15n + -1n2 = 0 + n2 + -1n2 7 + -15n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 7 + -15n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -7 + 15n + n2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 15n + 7 + n2 = 0 + 7 Reorder the terms: -7 + 7 + 15n + n2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 15n + n2 = 0 + 7 15n + n2 = 0 + 7 Combine like terms: 0 + 7 = 7 15n + n2 = 7 The n term is 15n. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15n + 56.25 + n2 = 7 + 56.25 Reorder the terms: 56.25 + 15n + n2 = 7 + 56.25 Combine like terms: 7 + 56.25 = 63.25 56.25 + 15n + n2 = 63.25 Factor a perfect square on the left side: (n + 7.5)(n + 7.5) = 63.25 Calculate the square root of the right side: 7.95298686 Break this problem into two subproblems by setting (n + 7.5) equal to 7.95298686 and -7.95298686.Subproblem 1
n + 7.5 = 7.95298686 Simplifying n + 7.5 = 7.95298686 Reorder the terms: 7.5 + n = 7.95298686 Solving 7.5 + n = 7.95298686 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + n = 7.95298686 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + n = 7.95298686 + -7.5 n = 7.95298686 + -7.5 Combine like terms: 7.95298686 + -7.5 = 0.45298686 n = 0.45298686 Simplifying n = 0.45298686Subproblem 2
n + 7.5 = -7.95298686 Simplifying n + 7.5 = -7.95298686 Reorder the terms: 7.5 + n = -7.95298686 Solving 7.5 + n = -7.95298686 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + n = -7.95298686 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + n = -7.95298686 + -7.5 n = -7.95298686 + -7.5 Combine like terms: -7.95298686 + -7.5 = -15.45298686 n = -15.45298686 Simplifying n = -15.45298686Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.45298686, -15.45298686}
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