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Simplifying 11n2 + -3n + -21 = 0 Reorder the terms: -21 + -3n + 11n2 = 0 Solving -21 + -3n + 11n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -1.909090909 + -0.2727272727n + n2 = 0 Move the constant term to the right: Add '1.909090909' to each side of the equation. -1.909090909 + -0.2727272727n + 1.909090909 + n2 = 0 + 1.909090909 Reorder the terms: -1.909090909 + 1.909090909 + -0.2727272727n + n2 = 0 + 1.909090909 Combine like terms: -1.909090909 + 1.909090909 = 0.000000000 0.000000000 + -0.2727272727n + n2 = 0 + 1.909090909 -0.2727272727n + n2 = 0 + 1.909090909 Combine like terms: 0 + 1.909090909 = 1.909090909 -0.2727272727n + n2 = 1.909090909 The n term is -0.2727272727n. Take half its coefficient (-0.1363636364). Square it (0.01859504133) and add it to both sides. Add '0.01859504133' to each side of the equation. -0.2727272727n + 0.01859504133 + n2 = 1.909090909 + 0.01859504133 Reorder the terms: 0.01859504133 + -0.2727272727n + n2 = 1.909090909 + 0.01859504133 Combine like terms: 1.909090909 + 0.01859504133 = 1.92768595033 0.01859504133 + -0.2727272727n + n2 = 1.92768595033 Factor a perfect square on the left side: (n + -0.1363636364)(n + -0.1363636364) = 1.92768595033 Calculate the square root of the right side: 1.388411304 Break this problem into two subproblems by setting (n + -0.1363636364) equal to 1.388411304 and -1.388411304.Subproblem 1
n + -0.1363636364 = 1.388411304 Simplifying n + -0.1363636364 = 1.388411304 Reorder the terms: -0.1363636364 + n = 1.388411304 Solving -0.1363636364 + n = 1.388411304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1363636364' to each side of the equation. -0.1363636364 + 0.1363636364 + n = 1.388411304 + 0.1363636364 Combine like terms: -0.1363636364 + 0.1363636364 = 0.0000000000 0.0000000000 + n = 1.388411304 + 0.1363636364 n = 1.388411304 + 0.1363636364 Combine like terms: 1.388411304 + 0.1363636364 = 1.5247749404 n = 1.5247749404 Simplifying n = 1.5247749404Subproblem 2
n + -0.1363636364 = -1.388411304 Simplifying n + -0.1363636364 = -1.388411304 Reorder the terms: -0.1363636364 + n = -1.388411304 Solving -0.1363636364 + n = -1.388411304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1363636364' to each side of the equation. -0.1363636364 + 0.1363636364 + n = -1.388411304 + 0.1363636364 Combine like terms: -0.1363636364 + 0.1363636364 = 0.0000000000 0.0000000000 + n = -1.388411304 + 0.1363636364 n = -1.388411304 + 0.1363636364 Combine like terms: -1.388411304 + 0.1363636364 = -1.2520476676 n = -1.2520476676 Simplifying n = -1.2520476676Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.5247749404, -1.2520476676}
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