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Simplifying 4n2 + 2n = 21 Reorder the terms: 2n + 4n2 = 21 Solving 2n + 4n2 = 21 Solving for variable 'n'. Reorder the terms: -21 + 2n + 4n2 = 21 + -21 Combine like terms: 21 + -21 = 0 -21 + 2n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5.25 + 0.5n + n2 = 0 Move the constant term to the right: Add '5.25' to each side of the equation. -5.25 + 0.5n + 5.25 + n2 = 0 + 5.25 Reorder the terms: -5.25 + 5.25 + 0.5n + n2 = 0 + 5.25 Combine like terms: -5.25 + 5.25 = 0.00 0.00 + 0.5n + n2 = 0 + 5.25 0.5n + n2 = 0 + 5.25 Combine like terms: 0 + 5.25 = 5.25 0.5n + n2 = 5.25 The n term is 0.5n. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5n + 0.0625 + n2 = 5.25 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 5.25 + 0.0625 Combine like terms: 5.25 + 0.0625 = 5.3125 0.0625 + 0.5n + n2 = 5.3125 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 5.3125 Calculate the square root of the right side: 2.304886114 Break this problem into two subproblems by setting (n + 0.25) equal to 2.304886114 and -2.304886114.Subproblem 1
n + 0.25 = 2.304886114 Simplifying n + 0.25 = 2.304886114 Reorder the terms: 0.25 + n = 2.304886114 Solving 0.25 + n = 2.304886114 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 2.304886114 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 2.304886114 + -0.25 n = 2.304886114 + -0.25 Combine like terms: 2.304886114 + -0.25 = 2.054886114 n = 2.054886114 Simplifying n = 2.054886114Subproblem 2
n + 0.25 = -2.304886114 Simplifying n + 0.25 = -2.304886114 Reorder the terms: 0.25 + n = -2.304886114 Solving 0.25 + n = -2.304886114 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -2.304886114 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -2.304886114 + -0.25 n = -2.304886114 + -0.25 Combine like terms: -2.304886114 + -0.25 = -2.554886114 n = -2.554886114 Simplifying n = -2.554886114Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.054886114, -2.554886114}
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