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Simplifying 8n2 + -3n + 10 + -3n2 + -6n + -7 = 0 Reorder the terms: 10 + -7 + -3n + -6n + 8n2 + -3n2 = 0 Combine like terms: 10 + -7 = 3 3 + -3n + -6n + 8n2 + -3n2 = 0 Combine like terms: -3n + -6n = -9n 3 + -9n + 8n2 + -3n2 = 0 Combine like terms: 8n2 + -3n2 = 5n2 3 + -9n + 5n2 = 0 Solving 3 + -9n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.6 + -1.8n + n2 = 0 Move the constant term to the right: Add '-0.6' to each side of the equation. 0.6 + -1.8n + -0.6 + n2 = 0 + -0.6 Reorder the terms: 0.6 + -0.6 + -1.8n + n2 = 0 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + -1.8n + n2 = 0 + -0.6 -1.8n + n2 = 0 + -0.6 Combine like terms: 0 + -0.6 = -0.6 -1.8n + n2 = -0.6 The n term is -1.8n. Take half its coefficient (-0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. -1.8n + 0.81 + n2 = -0.6 + 0.81 Reorder the terms: 0.81 + -1.8n + n2 = -0.6 + 0.81 Combine like terms: -0.6 + 0.81 = 0.21 0.81 + -1.8n + n2 = 0.21 Factor a perfect square on the left side: (n + -0.9)(n + -0.9) = 0.21 Calculate the square root of the right side: 0.458257569 Break this problem into two subproblems by setting (n + -0.9) equal to 0.458257569 and -0.458257569.Subproblem 1
n + -0.9 = 0.458257569 Simplifying n + -0.9 = 0.458257569 Reorder the terms: -0.9 + n = 0.458257569 Solving -0.9 + n = 0.458257569 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + n = 0.458257569 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + n = 0.458257569 + 0.9 n = 0.458257569 + 0.9 Combine like terms: 0.458257569 + 0.9 = 1.358257569 n = 1.358257569 Simplifying n = 1.358257569Subproblem 2
n + -0.9 = -0.458257569 Simplifying n + -0.9 = -0.458257569 Reorder the terms: -0.9 + n = -0.458257569 Solving -0.9 + n = -0.458257569 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + n = -0.458257569 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + n = -0.458257569 + 0.9 n = -0.458257569 + 0.9 Combine like terms: -0.458257569 + 0.9 = 0.441742431 n = 0.441742431 Simplifying n = 0.441742431Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.358257569, 0.441742431}
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