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Simplifying 4n2 + 9n + -3 = 0 Reorder the terms: -3 + 9n + 4n2 = 0 Solving -3 + 9n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 2.25n + n2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 2.25n + 0.75 + n2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 2.25n + n2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 2.25n + n2 = 0 + 0.75 2.25n + n2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 2.25n + n2 = 0.75 The n term is 2.25n. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25n + 1.265625 + n2 = 0.75 + 1.265625 Reorder the terms: 1.265625 + 2.25n + n2 = 0.75 + 1.265625 Combine like terms: 0.75 + 1.265625 = 2.015625 1.265625 + 2.25n + n2 = 2.015625 Factor a perfect square on the left side: (n + 1.125)(n + 1.125) = 2.015625 Calculate the square root of the right side: 1.419727086 Break this problem into two subproblems by setting (n + 1.125) equal to 1.419727086 and -1.419727086.Subproblem 1
n + 1.125 = 1.419727086 Simplifying n + 1.125 = 1.419727086 Reorder the terms: 1.125 + n = 1.419727086 Solving 1.125 + n = 1.419727086 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + n = 1.419727086 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + n = 1.419727086 + -1.125 n = 1.419727086 + -1.125 Combine like terms: 1.419727086 + -1.125 = 0.294727086 n = 0.294727086 Simplifying n = 0.294727086Subproblem 2
n + 1.125 = -1.419727086 Simplifying n + 1.125 = -1.419727086 Reorder the terms: 1.125 + n = -1.419727086 Solving 1.125 + n = -1.419727086 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + n = -1.419727086 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + n = -1.419727086 + -1.125 n = -1.419727086 + -1.125 Combine like terms: -1.419727086 + -1.125 = -2.544727086 n = -2.544727086 Simplifying n = -2.544727086Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.294727086, -2.544727086}
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