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Simplifying 4 + 9n + -3n2 = 2 + -1n2 Solving 4 + 9n + -3n2 = 2 + -1n2 Solving for variable 'n'. Reorder the terms: 4 + -2 + 9n + -3n2 + n2 = 2 + -1n2 + -2 + n2 Combine like terms: 4 + -2 = 2 2 + 9n + -3n2 + n2 = 2 + -1n2 + -2 + n2 Combine like terms: -3n2 + n2 = -2n2 2 + 9n + -2n2 = 2 + -1n2 + -2 + n2 Reorder the terms: 2 + 9n + -2n2 = 2 + -2 + -1n2 + n2 Combine like terms: 2 + -2 = 0 2 + 9n + -2n2 = 0 + -1n2 + n2 2 + 9n + -2n2 = -1n2 + n2 Combine like terms: -1n2 + n2 = 0 2 + 9n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1 + -4.5n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -4.5n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -4.5n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -4.5n + n2 = 0 + 1 -4.5n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -4.5n + n2 = 1 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 = 1 + 5.0625 Reorder the terms: 5.0625 + -4.5n + n2 = 1 + 5.0625 Combine like terms: 1 + 5.0625 = 6.0625 5.0625 + -4.5n + n2 = 6.0625 Factor a perfect square on the left side: (n + -2.25)(n + -2.25) = 6.0625 Calculate the square root of the right side: 2.46221445 Break this problem into two subproblems by setting (n + -2.25) equal to 2.46221445 and -2.46221445.Subproblem 1
n + -2.25 = 2.46221445 Simplifying n + -2.25 = 2.46221445 Reorder the terms: -2.25 + n = 2.46221445 Solving -2.25 + n = 2.46221445 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 = 2.46221445 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = 2.46221445 + 2.25 n = 2.46221445 + 2.25 Combine like terms: 2.46221445 + 2.25 = 4.71221445 n = 4.71221445 Simplifying n = 4.71221445Subproblem 2
n + -2.25 = -2.46221445 Simplifying n + -2.25 = -2.46221445 Reorder the terms: -2.25 + n = -2.46221445 Solving -2.25 + n = -2.46221445 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 = -2.46221445 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = -2.46221445 + 2.25 n = -2.46221445 + 2.25 Combine like terms: -2.46221445 + 2.25 = -0.21221445 n = -0.21221445 Simplifying n = -0.21221445Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.71221445, -0.21221445}
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