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Simplifying 4q2 + -9q + 3 = 0 Reorder the terms: 3 + -9q + 4q2 = 0 Solving 3 + -9q + 4q2 = 0 Solving for variable 'q'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + -2.25q + q2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2.25q + -0.75 + q2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2.25q + q2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2.25q + q2 = 0 + -0.75 -2.25q + q2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2.25q + q2 = -0.75 The q term is -2.25q. 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. -2.25q + 1.265625 + q2 = -0.75 + 1.265625 Reorder the terms: 1.265625 + -2.25q + q2 = -0.75 + 1.265625 Combine like terms: -0.75 + 1.265625 = 0.515625 1.265625 + -2.25q + q2 = 0.515625 Factor a perfect square on the left side: (q + -1.125)(q + -1.125) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (q + -1.125) equal to 0.718070331 and -0.718070331.Subproblem 1
q + -1.125 = 0.718070331 Simplifying q + -1.125 = 0.718070331 Reorder the terms: -1.125 + q = 0.718070331 Solving -1.125 + q = 0.718070331 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + q = 0.718070331 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + q = 0.718070331 + 1.125 q = 0.718070331 + 1.125 Combine like terms: 0.718070331 + 1.125 = 1.843070331 q = 1.843070331 Simplifying q = 1.843070331Subproblem 2
q + -1.125 = -0.718070331 Simplifying q + -1.125 = -0.718070331 Reorder the terms: -1.125 + q = -0.718070331 Solving -1.125 + q = -0.718070331 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + q = -0.718070331 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + q = -0.718070331 + 1.125 q = -0.718070331 + 1.125 Combine like terms: -0.718070331 + 1.125 = 0.406929669 q = 0.406929669 Simplifying q = 0.406929669Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.843070331, 0.406929669}
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