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Simplifying 0.5n2 + n = 6 Reorder the terms: n + 0.5n2 = 6 Solving n + 0.5n2 = 6 Solving for variable 'n'. Reorder the terms: -6 + n + 0.5n2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + n + 0.5n2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -12 + 2n + n2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 2n + 12 + n2 = 0 + 12 Reorder the terms: -12 + 12 + 2n + n2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 2n + n2 = 0 + 12 2n + n2 = 0 + 12 Combine like terms: 0 + 12 = 12 2n + n2 = 12 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 2n + 0.25 + n2 = 12 + 0.25 Reorder the terms: 0.25 + 2n + n2 = 12 + 0.25 Combine like terms: 12 + 0.25 = 12.25 0.25 + 2n + n2 = 12.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 12.25 Calculate the square root of the right side: 3.5 Break this problem into two subproblems by setting (n + 0.5) equal to 3.5 and -3.5.Subproblem 1
n + 0.5 = 3.5 Simplifying n + 0.5 = 3.5 Reorder the terms: 0.5 + n = 3.5 Solving 0.5 + n = 3.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 3.5 + -0.5 n = 3.5 + -0.5 Combine like terms: 3.5 + -0.5 = 3 n = 3 Simplifying n = 3Subproblem 2
n + 0.5 = -3.5 Simplifying n + 0.5 = -3.5 Reorder the terms: 0.5 + n = -3.5 Solving 0.5 + n = -3.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -3.5 + -0.5 n = -3.5 + -0.5 Combine like terms: -3.5 + -0.5 = -4 n = -4 Simplifying n = -4Solution
The solution to the problem is based on the solutions from the subproblems. n = {3, -4}
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