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Simplifying 6q2 + -5q = 5q + -7q + 48 Reorder the terms: -5q + 6q2 = 5q + -7q + 48 Reorder the terms: -5q + 6q2 = 48 + 5q + -7q Combine like terms: 5q + -7q = -2q -5q + 6q2 = 48 + -2q Solving -5q + 6q2 = 48 + -2q Solving for variable 'q'. Reorder the terms: -48 + -5q + 2q + 6q2 = 48 + -2q + -48 + 2q Combine like terms: -5q + 2q = -3q -48 + -3q + 6q2 = 48 + -2q + -48 + 2q Reorder the terms: -48 + -3q + 6q2 = 48 + -48 + -2q + 2q Combine like terms: 48 + -48 = 0 -48 + -3q + 6q2 = 0 + -2q + 2q -48 + -3q + 6q2 = -2q + 2q Combine like terms: -2q + 2q = 0 -48 + -3q + 6q2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-16 + -1q + 2q2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-16 + -1q + 2q2)' equal to zero and attempt to solve: Simplifying -16 + -1q + 2q2 = 0 Solving -16 + -1q + 2q2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8 + -0.5q + q2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -0.5q + 8 + q2 = 0 + 8 Reorder the terms: -8 + 8 + -0.5q + q2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -0.5q + q2 = 0 + 8 -0.5q + q2 = 0 + 8 Combine like terms: 0 + 8 = 8 -0.5q + q2 = 8 The q term is -0.5q. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5q + 0.0625 + q2 = 8 + 0.0625 Reorder the terms: 0.0625 + -0.5q + q2 = 8 + 0.0625 Combine like terms: 8 + 0.0625 = 8.0625 0.0625 + -0.5q + q2 = 8.0625 Factor a perfect square on the left side: (q + -0.25)(q + -0.25) = 8.0625 Calculate the square root of the right side: 2.839454173 Break this problem into two subproblems by setting (q + -0.25) equal to 2.839454173 and -2.839454173.Subproblem 1
q + -0.25 = 2.839454173 Simplifying q + -0.25 = 2.839454173 Reorder the terms: -0.25 + q = 2.839454173 Solving -0.25 + q = 2.839454173 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + q = 2.839454173 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + q = 2.839454173 + 0.25 q = 2.839454173 + 0.25 Combine like terms: 2.839454173 + 0.25 = 3.089454173 q = 3.089454173 Simplifying q = 3.089454173Subproblem 2
q + -0.25 = -2.839454173 Simplifying q + -0.25 = -2.839454173 Reorder the terms: -0.25 + q = -2.839454173 Solving -0.25 + q = -2.839454173 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + q = -2.839454173 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + q = -2.839454173 + 0.25 q = -2.839454173 + 0.25 Combine like terms: -2.839454173 + 0.25 = -2.589454173 q = -2.589454173 Simplifying q = -2.589454173Solution
The solution to the problem is based on the solutions from the subproblems. q = {3.089454173, -2.589454173}Solution
q = {3.089454173, -2.589454173}
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