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Simplifying 4n2 + 6n + 9 = 9n2 + 4n + 4 Reorder the terms: 9 + 6n + 4n2 = 9n2 + 4n + 4 Reorder the terms: 9 + 6n + 4n2 = 4 + 4n + 9n2 Solving 9 + 6n + 4n2 = 4 + 4n + 9n2 Solving for variable 'n'. Reorder the terms: 9 + -4 + 6n + -4n + 4n2 + -9n2 = 4 + 4n + 9n2 + -4 + -4n + -9n2 Combine like terms: 9 + -4 = 5 5 + 6n + -4n + 4n2 + -9n2 = 4 + 4n + 9n2 + -4 + -4n + -9n2 Combine like terms: 6n + -4n = 2n 5 + 2n + 4n2 + -9n2 = 4 + 4n + 9n2 + -4 + -4n + -9n2 Combine like terms: 4n2 + -9n2 = -5n2 5 + 2n + -5n2 = 4 + 4n + 9n2 + -4 + -4n + -9n2 Reorder the terms: 5 + 2n + -5n2 = 4 + -4 + 4n + -4n + 9n2 + -9n2 Combine like terms: 4 + -4 = 0 5 + 2n + -5n2 = 0 + 4n + -4n + 9n2 + -9n2 5 + 2n + -5n2 = 4n + -4n + 9n2 + -9n2 Combine like terms: 4n + -4n = 0 5 + 2n + -5n2 = 0 + 9n2 + -9n2 5 + 2n + -5n2 = 9n2 + -9n2 Combine like terms: 9n2 + -9n2 = 0 5 + 2n + -5n2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -1 + -0.4n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.4n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -0.4n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.4n + n2 = 0 + 1 -0.4n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.4n + n2 = 1 The n term is -0.4n. Take half its coefficient (-0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. -0.4n + 0.04 + n2 = 1 + 0.04 Reorder the terms: 0.04 + -0.4n + n2 = 1 + 0.04 Combine like terms: 1 + 0.04 = 1.04 0.04 + -0.4n + n2 = 1.04 Factor a perfect square on the left side: (n + -0.2)(n + -0.2) = 1.04 Calculate the square root of the right side: 1.019803903 Break this problem into two subproblems by setting (n + -0.2) equal to 1.019803903 and -1.019803903.Subproblem 1
n + -0.2 = 1.019803903 Simplifying n + -0.2 = 1.019803903 Reorder the terms: -0.2 + n = 1.019803903 Solving -0.2 + n = 1.019803903 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + n = 1.019803903 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + n = 1.019803903 + 0.2 n = 1.019803903 + 0.2 Combine like terms: 1.019803903 + 0.2 = 1.219803903 n = 1.219803903 Simplifying n = 1.219803903Subproblem 2
n + -0.2 = -1.019803903 Simplifying n + -0.2 = -1.019803903 Reorder the terms: -0.2 + n = -1.019803903 Solving -0.2 + n = -1.019803903 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + n = -1.019803903 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + n = -1.019803903 + 0.2 n = -1.019803903 + 0.2 Combine like terms: -1.019803903 + 0.2 = -0.819803903 n = -0.819803903 Simplifying n = -0.819803903Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.219803903, -0.819803903}
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