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Simplifying 9n2 = 4n + 9 Reorder the terms: 9n2 = 9 + 4n Solving 9n2 = 9 + 4n Solving for variable 'n'. Reorder the terms: -9 + -4n + 9n2 = 9 + 4n + -9 + -4n Reorder the terms: -9 + -4n + 9n2 = 9 + -9 + 4n + -4n Combine like terms: 9 + -9 = 0 -9 + -4n + 9n2 = 0 + 4n + -4n -9 + -4n + 9n2 = 4n + -4n Combine like terms: 4n + -4n = 0 -9 + -4n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1 + -0.4444444444n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.4444444444n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -0.4444444444n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.4444444444n + n2 = 0 + 1 -0.4444444444n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.4444444444n + n2 = 1 The n term is -0.4444444444n. Take half its coefficient (-0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. -0.4444444444n + 0.04938271604 + n2 = 1 + 0.04938271604 Reorder the terms: 0.04938271604 + -0.4444444444n + n2 = 1 + 0.04938271604 Combine like terms: 1 + 0.04938271604 = 1.04938271604 0.04938271604 + -0.4444444444n + n2 = 1.04938271604 Factor a perfect square on the left side: (n + -0.2222222222)(n + -0.2222222222) = 1.04938271604 Calculate the square root of the right side: 1.024393829 Break this problem into two subproblems by setting (n + -0.2222222222) equal to 1.024393829 and -1.024393829.Subproblem 1
n + -0.2222222222 = 1.024393829 Simplifying n + -0.2222222222 = 1.024393829 Reorder the terms: -0.2222222222 + n = 1.024393829 Solving -0.2222222222 + n = 1.024393829 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + n = 1.024393829 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + n = 1.024393829 + 0.2222222222 n = 1.024393829 + 0.2222222222 Combine like terms: 1.024393829 + 0.2222222222 = 1.2466160512 n = 1.2466160512 Simplifying n = 1.2466160512Subproblem 2
n + -0.2222222222 = -1.024393829 Simplifying n + -0.2222222222 = -1.024393829 Reorder the terms: -0.2222222222 + n = -1.024393829 Solving -0.2222222222 + n = -1.024393829 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + n = -1.024393829 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + n = -1.024393829 + 0.2222222222 n = -1.024393829 + 0.2222222222 Combine like terms: -1.024393829 + 0.2222222222 = -0.8021716068 n = -0.8021716068 Simplifying n = -0.8021716068Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.2466160512, -0.8021716068}
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