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Simplifying 2n2 + -9n = 9 Reorder the terms: -9n + 2n2 = 9 Solving -9n + 2n2 = 9 Solving for variable 'n'. Reorder the terms: -9 + -9n + 2n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + -9n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + -4.5n + n2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + -4.5n + 4.5 + n2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + -4.5n + n2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + -4.5n + n2 = 0 + 4.5 -4.5n + n2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 -4.5n + n2 = 4.5 The n term is -4.5n. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5n + 5.0625 + n2 = 4.5 + 5.0625 Reorder the terms: 5.0625 + -4.5n + n2 = 4.5 + 5.0625 Combine like terms: 4.5 + 5.0625 = 9.5625 5.0625 + -4.5n + n2 = 9.5625 Factor a perfect square on the left side: (n + -2.25)(n + -2.25) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (n + -2.25) equal to 3.092329219 and -3.092329219.Subproblem 1
n + -2.25 = 3.092329219 Simplifying n + -2.25 = 3.092329219 Reorder the terms: -2.25 + n = 3.092329219 Solving -2.25 + n = 3.092329219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + n = 3.092329219 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = 3.092329219 + 2.25 n = 3.092329219 + 2.25 Combine like terms: 3.092329219 + 2.25 = 5.342329219 n = 5.342329219 Simplifying n = 5.342329219Subproblem 2
n + -2.25 = -3.092329219 Simplifying n + -2.25 = -3.092329219 Reorder the terms: -2.25 + n = -3.092329219 Solving -2.25 + n = -3.092329219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + n = -3.092329219 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = -3.092329219 + 2.25 n = -3.092329219 + 2.25 Combine like terms: -3.092329219 + 2.25 = -0.842329219 n = -0.842329219 Simplifying n = -0.842329219Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.342329219, -0.842329219}
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