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Simplifying 0 = 4q2 + 12q + -9 Reorder the terms: 0 = -9 + 12q + 4q2 Solving 0 = -9 + 12q + 4q2 Solving for variable 'q'. Combine like terms: 0 + 9 = 9 9 + -12q + -4q2 = -9 + 12q + 4q2 + 9 + -12q + -4q2 Reorder the terms: 9 + -12q + -4q2 = -9 + 9 + 12q + -12q + 4q2 + -4q2 Combine like terms: -9 + 9 = 0 9 + -12q + -4q2 = 0 + 12q + -12q + 4q2 + -4q2 9 + -12q + -4q2 = 12q + -12q + 4q2 + -4q2 Combine like terms: 12q + -12q = 0 9 + -12q + -4q2 = 0 + 4q2 + -4q2 9 + -12q + -4q2 = 4q2 + -4q2 Combine like terms: 4q2 + -4q2 = 0 9 + -12q + -4q2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.25 + 3q + q2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + 3q + 2.25 + q2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + 3q + q2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + 3q + q2 = 0 + 2.25 3q + q2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 3q + q2 = 2.25 The q term is 3q. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3q + 2.25 + q2 = 2.25 + 2.25 Reorder the terms: 2.25 + 3q + q2 = 2.25 + 2.25 Combine like terms: 2.25 + 2.25 = 4.5 2.25 + 3q + q2 = 4.5 Factor a perfect square on the left side: (q + 1.5)(q + 1.5) = 4.5 Calculate the square root of the right side: 2.121320344 Break this problem into two subproblems by setting (q + 1.5) equal to 2.121320344 and -2.121320344.Subproblem 1
q + 1.5 = 2.121320344 Simplifying q + 1.5 = 2.121320344 Reorder the terms: 1.5 + q = 2.121320344 Solving 1.5 + q = 2.121320344 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + q = 2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + q = 2.121320344 + -1.5 q = 2.121320344 + -1.5 Combine like terms: 2.121320344 + -1.5 = 0.621320344 q = 0.621320344 Simplifying q = 0.621320344Subproblem 2
q + 1.5 = -2.121320344 Simplifying q + 1.5 = -2.121320344 Reorder the terms: 1.5 + q = -2.121320344 Solving 1.5 + q = -2.121320344 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + q = -2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + q = -2.121320344 + -1.5 q = -2.121320344 + -1.5 Combine like terms: -2.121320344 + -1.5 = -3.621320344 q = -3.621320344 Simplifying q = -3.621320344Solution
The solution to the problem is based on the solutions from the subproblems. q = {0.621320344, -3.621320344}
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