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Simplifying 0.25q2 + -6q + 40 = 0.5q2 + -7q + 35 Reorder the terms: 40 + -6q + 0.25q2 = 0.5q2 + -7q + 35 Reorder the terms: 40 + -6q + 0.25q2 = 35 + -7q + 0.5q2 Solving 40 + -6q + 0.25q2 = 35 + -7q + 0.5q2 Solving for variable 'q'. Reorder the terms: 40 + -35 + -6q + 7q + 0.25q2 + -0.5q2 = 35 + -7q + 0.5q2 + -35 + 7q + -0.5q2 Combine like terms: 40 + -35 = 5 5 + -6q + 7q + 0.25q2 + -0.5q2 = 35 + -7q + 0.5q2 + -35 + 7q + -0.5q2 Combine like terms: -6q + 7q = 1q 5 + 1q + 0.25q2 + -0.5q2 = 35 + -7q + 0.5q2 + -35 + 7q + -0.5q2 Combine like terms: 0.25q2 + -0.5q2 = -0.25q2 5 + 1q + -0.25q2 = 35 + -7q + 0.5q2 + -35 + 7q + -0.5q2 Reorder the terms: 5 + 1q + -0.25q2 = 35 + -35 + -7q + 7q + 0.5q2 + -0.5q2 Combine like terms: 35 + -35 = 0 5 + 1q + -0.25q2 = 0 + -7q + 7q + 0.5q2 + -0.5q2 5 + 1q + -0.25q2 = -7q + 7q + 0.5q2 + -0.5q2 Combine like terms: -7q + 7q = 0 5 + 1q + -0.25q2 = 0 + 0.5q2 + -0.5q2 5 + 1q + -0.25q2 = 0.5q2 + -0.5q2 Combine like terms: 0.5q2 + -0.5q2 = 0.0 5 + 1q + -0.25q2 = 0.0 Begin completing the square. Divide all terms by -0.25 the coefficient of the squared term: Divide each side by '-0.25'. -20 + -4q + q2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + -4q + 20 + q2 = 0 + 20 Reorder the terms: -20 + 20 + -4q + q2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + -4q + q2 = 0 + 20 -4q + q2 = 0 + 20 Combine like terms: 0 + 20 = 20 -4q + q2 = 20 The q term is -4q. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4q + 4 + q2 = 20 + 4 Reorder the terms: 4 + -4q + q2 = 20 + 4 Combine like terms: 20 + 4 = 24 4 + -4q + q2 = 24 Factor a perfect square on the left side: (q + -2)(q + -2) = 24 Calculate the square root of the right side: 4.898979486 Break this problem into two subproblems by setting (q + -2) equal to 4.898979486 and -4.898979486.Subproblem 1
q + -2 = 4.898979486 Simplifying q + -2 = 4.898979486 Reorder the terms: -2 + q = 4.898979486 Solving -2 + q = 4.898979486 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + q = 4.898979486 + 2 Combine like terms: -2 + 2 = 0 0 + q = 4.898979486 + 2 q = 4.898979486 + 2 Combine like terms: 4.898979486 + 2 = 6.898979486 q = 6.898979486 Simplifying q = 6.898979486Subproblem 2
q + -2 = -4.898979486 Simplifying q + -2 = -4.898979486 Reorder the terms: -2 + q = -4.898979486 Solving -2 + q = -4.898979486 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + q = -4.898979486 + 2 Combine like terms: -2 + 2 = 0 0 + q = -4.898979486 + 2 q = -4.898979486 + 2 Combine like terms: -4.898979486 + 2 = -2.898979486 q = -2.898979486 Simplifying q = -2.898979486Solution
The solution to the problem is based on the solutions from the subproblems. q = {6.898979486, -2.898979486}
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