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Simplifying 2n2 = 21 + 7n Solving 2n2 = 21 + 7n Solving for variable 'n'. Reorder the terms: -21 + -7n + 2n2 = 21 + 7n + -21 + -7n Reorder the terms: -21 + -7n + 2n2 = 21 + -21 + 7n + -7n Combine like terms: 21 + -21 = 0 -21 + -7n + 2n2 = 0 + 7n + -7n -21 + -7n + 2n2 = 7n + -7n Combine like terms: 7n + -7n = 0 -21 + -7n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -10.5 + -3.5n + n2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + -3.5n + 10.5 + n2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + -3.5n + n2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + -3.5n + n2 = 0 + 10.5 -3.5n + n2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 -3.5n + n2 = 10.5 The n term is -3.5n. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5n + 3.0625 + n2 = 10.5 + 3.0625 Reorder the terms: 3.0625 + -3.5n + n2 = 10.5 + 3.0625 Combine like terms: 10.5 + 3.0625 = 13.5625 3.0625 + -3.5n + n2 = 13.5625 Factor a perfect square on the left side: (n + -1.75)(n + -1.75) = 13.5625 Calculate the square root of the right side: 3.682729966 Break this problem into two subproblems by setting (n + -1.75) equal to 3.682729966 and -3.682729966.Subproblem 1
n + -1.75 = 3.682729966 Simplifying n + -1.75 = 3.682729966 Reorder the terms: -1.75 + n = 3.682729966 Solving -1.75 + n = 3.682729966 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = 3.682729966 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = 3.682729966 + 1.75 n = 3.682729966 + 1.75 Combine like terms: 3.682729966 + 1.75 = 5.432729966 n = 5.432729966 Simplifying n = 5.432729966Subproblem 2
n + -1.75 = -3.682729966 Simplifying n + -1.75 = -3.682729966 Reorder the terms: -1.75 + n = -3.682729966 Solving -1.75 + n = -3.682729966 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = -3.682729966 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = -3.682729966 + 1.75 n = -3.682729966 + 1.75 Combine like terms: -3.682729966 + 1.75 = -1.932729966 n = -1.932729966 Simplifying n = -1.932729966Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.432729966, -1.932729966}
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