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Simplifying 9n2 + -2n = 10 Reorder the terms: -2n + 9n2 = 10 Solving -2n + 9n2 = 10 Solving for variable 'n'. Reorder the terms: -10 + -2n + 9n2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + -2n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.111111111 + -0.2222222222n + n2 = 0 Move the constant term to the right: Add '1.111111111' to each side of the equation. -1.111111111 + -0.2222222222n + 1.111111111 + n2 = 0 + 1.111111111 Reorder the terms: -1.111111111 + 1.111111111 + -0.2222222222n + n2 = 0 + 1.111111111 Combine like terms: -1.111111111 + 1.111111111 = 0.000000000 0.000000000 + -0.2222222222n + n2 = 0 + 1.111111111 -0.2222222222n + n2 = 0 + 1.111111111 Combine like terms: 0 + 1.111111111 = 1.111111111 -0.2222222222n + n2 = 1.111111111 The n term is -0.2222222222n. Take half its coefficient (-0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. -0.2222222222n + 0.01234567901 + n2 = 1.111111111 + 0.01234567901 Reorder the terms: 0.01234567901 + -0.2222222222n + n2 = 1.111111111 + 0.01234567901 Combine like terms: 1.111111111 + 0.01234567901 = 1.12345679001 0.01234567901 + -0.2222222222n + n2 = 1.12345679001 Factor a perfect square on the left side: (n + -0.1111111111)(n + -0.1111111111) = 1.12345679001 Calculate the square root of the right side: 1.059932446 Break this problem into two subproblems by setting (n + -0.1111111111) equal to 1.059932446 and -1.059932446.Subproblem 1
n + -0.1111111111 = 1.059932446 Simplifying n + -0.1111111111 = 1.059932446 Reorder the terms: -0.1111111111 + n = 1.059932446 Solving -0.1111111111 + n = 1.059932446 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.1111111111 + n = 1.059932446 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + n = 1.059932446 + 0.1111111111 n = 1.059932446 + 0.1111111111 Combine like terms: 1.059932446 + 0.1111111111 = 1.1710435571 n = 1.1710435571 Simplifying n = 1.1710435571Subproblem 2
n + -0.1111111111 = -1.059932446 Simplifying n + -0.1111111111 = -1.059932446 Reorder the terms: -0.1111111111 + n = -1.059932446 Solving -0.1111111111 + n = -1.059932446 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.1111111111 + n = -1.059932446 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + n = -1.059932446 + 0.1111111111 n = -1.059932446 + 0.1111111111 Combine like terms: -1.059932446 + 0.1111111111 = -0.9488213349 n = -0.9488213349 Simplifying n = -0.9488213349Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.1710435571, -0.9488213349}
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