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Simplifying 0.5n2 + -1.5n + -77 = 0 Reorder the terms: -77 + -1.5n + 0.5n2 = 0 Solving -77 + -1.5n + 0.5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -154 + -3n + n2 = 0 Move the constant term to the right: Add '154' to each side of the equation. -154 + -3n + 154 + n2 = 0 + 154 Reorder the terms: -154 + 154 + -3n + n2 = 0 + 154 Combine like terms: -154 + 154 = 0 0 + -3n + n2 = 0 + 154 -3n + n2 = 0 + 154 Combine like terms: 0 + 154 = 154 -3n + n2 = 154 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 154 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 154 + 2.25 Combine like terms: 154 + 2.25 = 156.25 2.25 + -3n + n2 = 156.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 156.25 Calculate the square root of the right side: 12.5 Break this problem into two subproblems by setting (n + -1.5) equal to 12.5 and -12.5.Subproblem 1
n + -1.5 = 12.5 Simplifying n + -1.5 = 12.5 Reorder the terms: -1.5 + n = 12.5 Solving -1.5 + n = 12.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 12.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 12.5 + 1.5 n = 12.5 + 1.5 Combine like terms: 12.5 + 1.5 = 14 n = 14 Simplifying n = 14Subproblem 2
n + -1.5 = -12.5 Simplifying n + -1.5 = -12.5 Reorder the terms: -1.5 + n = -12.5 Solving -1.5 + n = -12.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -12.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -12.5 + 1.5 n = -12.5 + 1.5 Combine like terms: -12.5 + 1.5 = -11 n = -11 Simplifying n = -11Solution
The solution to the problem is based on the solutions from the subproblems. n = {14, -11}
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