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Simplifying 8n2 + 4n + -21 = 0 Reorder the terms: -21 + 4n + 8n2 = 0 Solving -21 + 4n + 8n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.625 + 0.5n + n2 = 0 Move the constant term to the right: Add '2.625' to each side of the equation. -2.625 + 0.5n + 2.625 + n2 = 0 + 2.625 Reorder the terms: -2.625 + 2.625 + 0.5n + n2 = 0 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + 0.5n + n2 = 0 + 2.625 0.5n + n2 = 0 + 2.625 Combine like terms: 0 + 2.625 = 2.625 0.5n + n2 = 2.625 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 = 2.625 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 2.625 + 0.0625 Combine like terms: 2.625 + 0.0625 = 2.6875 0.0625 + 0.5n + n2 = 2.6875 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 2.6875 Calculate the square root of the right side: 1.639359631 Break this problem into two subproblems by setting (n + 0.25) equal to 1.639359631 and -1.639359631.Subproblem 1
n + 0.25 = 1.639359631 Simplifying n + 0.25 = 1.639359631 Reorder the terms: 0.25 + n = 1.639359631 Solving 0.25 + n = 1.639359631 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 = 1.639359631 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.639359631 + -0.25 n = 1.639359631 + -0.25 Combine like terms: 1.639359631 + -0.25 = 1.389359631 n = 1.389359631 Simplifying n = 1.389359631Subproblem 2
n + 0.25 = -1.639359631 Simplifying n + 0.25 = -1.639359631 Reorder the terms: 0.25 + n = -1.639359631 Solving 0.25 + n = -1.639359631 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 = -1.639359631 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.639359631 + -0.25 n = -1.639359631 + -0.25 Combine like terms: -1.639359631 + -0.25 = -1.889359631 n = -1.889359631 Simplifying n = -1.889359631Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.389359631, -1.889359631}
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