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Simplifying 90n2 + 85n + -21090 = 0 Reorder the terms: -21090 + 85n + 90n2 = 0 Solving -21090 + 85n + 90n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '5'. 5(-4218 + 17n + 18n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-4218 + 17n + 18n2)' equal to zero and attempt to solve: Simplifying -4218 + 17n + 18n2 = 0 Solving -4218 + 17n + 18n2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -234.3333333 + 0.9444444444n + n2 = 0 Move the constant term to the right: Add '234.3333333' to each side of the equation. -234.3333333 + 0.9444444444n + 234.3333333 + n2 = 0 + 234.3333333 Reorder the terms: -234.3333333 + 234.3333333 + 0.9444444444n + n2 = 0 + 234.3333333 Combine like terms: -234.3333333 + 234.3333333 = 0.0000000 0.0000000 + 0.9444444444n + n2 = 0 + 234.3333333 0.9444444444n + n2 = 0 + 234.3333333 Combine like terms: 0 + 234.3333333 = 234.3333333 0.9444444444n + n2 = 234.3333333 The n term is 0.9444444444n. Take half its coefficient (0.4722222222). Square it (0.2229938271) and add it to both sides. Add '0.2229938271' to each side of the equation. 0.9444444444n + 0.2229938271 + n2 = 234.3333333 + 0.2229938271 Reorder the terms: 0.2229938271 + 0.9444444444n + n2 = 234.3333333 + 0.2229938271 Combine like terms: 234.3333333 + 0.2229938271 = 234.5563271271 0.2229938271 + 0.9444444444n + n2 = 234.5563271271 Factor a perfect square on the left side: (n + 0.4722222222)(n + 0.4722222222) = 234.5563271271 Calculate the square root of the right side: 15.315231867 Break this problem into two subproblems by setting (n + 0.4722222222) equal to 15.315231867 and -15.315231867.Subproblem 1
n + 0.4722222222 = 15.315231867 Simplifying n + 0.4722222222 = 15.315231867 Reorder the terms: 0.4722222222 + n = 15.315231867 Solving 0.4722222222 + n = 15.315231867 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4722222222' to each side of the equation. 0.4722222222 + -0.4722222222 + n = 15.315231867 + -0.4722222222 Combine like terms: 0.4722222222 + -0.4722222222 = 0.0000000000 0.0000000000 + n = 15.315231867 + -0.4722222222 n = 15.315231867 + -0.4722222222 Combine like terms: 15.315231867 + -0.4722222222 = 14.8430096448 n = 14.8430096448 Simplifying n = 14.8430096448Subproblem 2
n + 0.4722222222 = -15.315231867 Simplifying n + 0.4722222222 = -15.315231867 Reorder the terms: 0.4722222222 + n = -15.315231867 Solving 0.4722222222 + n = -15.315231867 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4722222222' to each side of the equation. 0.4722222222 + -0.4722222222 + n = -15.315231867 + -0.4722222222 Combine like terms: 0.4722222222 + -0.4722222222 = 0.0000000000 0.0000000000 + n = -15.315231867 + -0.4722222222 n = -15.315231867 + -0.4722222222 Combine like terms: -15.315231867 + -0.4722222222 = -15.7874540892 n = -15.7874540892 Simplifying n = -15.7874540892Solution
The solution to the problem is based on the solutions from the subproblems. n = {14.8430096448, -15.7874540892}Solution
n = {14.8430096448, -15.7874540892}
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