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Simplifying 7.5n + -11 = n2 Reorder the terms: -11 + 7.5n = n2 Solving -11 + 7.5n = n2 Solving for variable 'n'. Combine like terms: n2 + -1n2 = 0 -11 + 7.5n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 11 + -7.5n + n2 = 0 Move the constant term to the right: Add '-11' to each side of the equation. 11 + -7.5n + -11 + n2 = 0 + -11 Reorder the terms: 11 + -11 + -7.5n + n2 = 0 + -11 Combine like terms: 11 + -11 = 0 0 + -7.5n + n2 = 0 + -11 -7.5n + n2 = 0 + -11 Combine like terms: 0 + -11 = -11 -7.5n + n2 = -11 The n term is -7.5n. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5n + 14.0625 + n2 = -11 + 14.0625 Reorder the terms: 14.0625 + -7.5n + n2 = -11 + 14.0625 Combine like terms: -11 + 14.0625 = 3.0625 14.0625 + -7.5n + n2 = 3.0625 Factor a perfect square on the left side: (n + -3.75)(n + -3.75) = 3.0625 Calculate the square root of the right side: 1.75 Break this problem into two subproblems by setting (n + -3.75) equal to 1.75 and -1.75.Subproblem 1
n + -3.75 = 1.75 Simplifying n + -3.75 = 1.75 Reorder the terms: -3.75 + n = 1.75 Solving -3.75 + n = 1.75 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + n = 1.75 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + n = 1.75 + 3.75 n = 1.75 + 3.75 Combine like terms: 1.75 + 3.75 = 5.5 n = 5.5 Simplifying n = 5.5Subproblem 2
n + -3.75 = -1.75 Simplifying n + -3.75 = -1.75 Reorder the terms: -3.75 + n = -1.75 Solving -3.75 + n = -1.75 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + n = -1.75 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + n = -1.75 + 3.75 n = -1.75 + 3.75 Combine like terms: -1.75 + 3.75 = 2 n = 2 Simplifying n = 2Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.5, 2}
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