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Simplifying q2 + 4q + 8 = 2q2 + -20q + 68 Reorder the terms: 8 + 4q + q2 = 2q2 + -20q + 68 Reorder the terms: 8 + 4q + q2 = 68 + -20q + 2q2 Solving 8 + 4q + q2 = 68 + -20q + 2q2 Solving for variable 'q'. Reorder the terms: 8 + -68 + 4q + 20q + q2 + -2q2 = 68 + -20q + 2q2 + -68 + 20q + -2q2 Combine like terms: 8 + -68 = -60 -60 + 4q + 20q + q2 + -2q2 = 68 + -20q + 2q2 + -68 + 20q + -2q2 Combine like terms: 4q + 20q = 24q -60 + 24q + q2 + -2q2 = 68 + -20q + 2q2 + -68 + 20q + -2q2 Combine like terms: q2 + -2q2 = -1q2 -60 + 24q + -1q2 = 68 + -20q + 2q2 + -68 + 20q + -2q2 Reorder the terms: -60 + 24q + -1q2 = 68 + -68 + -20q + 20q + 2q2 + -2q2 Combine like terms: 68 + -68 = 0 -60 + 24q + -1q2 = 0 + -20q + 20q + 2q2 + -2q2 -60 + 24q + -1q2 = -20q + 20q + 2q2 + -2q2 Combine like terms: -20q + 20q = 0 -60 + 24q + -1q2 = 0 + 2q2 + -2q2 -60 + 24q + -1q2 = 2q2 + -2q2 Combine like terms: 2q2 + -2q2 = 0 -60 + 24q + -1q2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 60 + -24q + q2 = 0 Move the constant term to the right: Add '-60' to each side of the equation. 60 + -24q + -60 + q2 = 0 + -60 Reorder the terms: 60 + -60 + -24q + q2 = 0 + -60 Combine like terms: 60 + -60 = 0 0 + -24q + q2 = 0 + -60 -24q + q2 = 0 + -60 Combine like terms: 0 + -60 = -60 -24q + q2 = -60 The q term is -24q. Take half its coefficient (-12). Square it (144) and add it to both sides. Add '144' to each side of the equation. -24q + 144 + q2 = -60 + 144 Reorder the terms: 144 + -24q + q2 = -60 + 144 Combine like terms: -60 + 144 = 84 144 + -24q + q2 = 84 Factor a perfect square on the left side: (q + -12)(q + -12) = 84 Calculate the square root of the right side: 9.16515139 Break this problem into two subproblems by setting (q + -12) equal to 9.16515139 and -9.16515139.Subproblem 1
q + -12 = 9.16515139 Simplifying q + -12 = 9.16515139 Reorder the terms: -12 + q = 9.16515139 Solving -12 + q = 9.16515139 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + q = 9.16515139 + 12 Combine like terms: -12 + 12 = 0 0 + q = 9.16515139 + 12 q = 9.16515139 + 12 Combine like terms: 9.16515139 + 12 = 21.16515139 q = 21.16515139 Simplifying q = 21.16515139Subproblem 2
q + -12 = -9.16515139 Simplifying q + -12 = -9.16515139 Reorder the terms: -12 + q = -9.16515139 Solving -12 + q = -9.16515139 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + q = -9.16515139 + 12 Combine like terms: -12 + 12 = 0 0 + q = -9.16515139 + 12 q = -9.16515139 + 12 Combine like terms: -9.16515139 + 12 = 2.83484861 q = 2.83484861 Simplifying q = 2.83484861Solution
The solution to the problem is based on the solutions from the subproblems. q = {21.16515139, 2.83484861}
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