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Simplifying 600 + -32q = q2 + -56q + 422 Reorder the terms: 600 + -32q = 422 + -56q + q2 Solving 600 + -32q = 422 + -56q + q2 Solving for variable 'q'. Reorder the terms: 600 + -422 + -32q + 56q + -1q2 = 422 + -56q + q2 + -422 + 56q + -1q2 Combine like terms: 600 + -422 = 178 178 + -32q + 56q + -1q2 = 422 + -56q + q2 + -422 + 56q + -1q2 Combine like terms: -32q + 56q = 24q 178 + 24q + -1q2 = 422 + -56q + q2 + -422 + 56q + -1q2 Reorder the terms: 178 + 24q + -1q2 = 422 + -422 + -56q + 56q + q2 + -1q2 Combine like terms: 422 + -422 = 0 178 + 24q + -1q2 = 0 + -56q + 56q + q2 + -1q2 178 + 24q + -1q2 = -56q + 56q + q2 + -1q2 Combine like terms: -56q + 56q = 0 178 + 24q + -1q2 = 0 + q2 + -1q2 178 + 24q + -1q2 = q2 + -1q2 Combine like terms: q2 + -1q2 = 0 178 + 24q + -1q2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -178 + -24q + q2 = 0 Move the constant term to the right: Add '178' to each side of the equation. -178 + -24q + 178 + q2 = 0 + 178 Reorder the terms: -178 + 178 + -24q + q2 = 0 + 178 Combine like terms: -178 + 178 = 0 0 + -24q + q2 = 0 + 178 -24q + q2 = 0 + 178 Combine like terms: 0 + 178 = 178 -24q + q2 = 178 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 = 178 + 144 Reorder the terms: 144 + -24q + q2 = 178 + 144 Combine like terms: 178 + 144 = 322 144 + -24q + q2 = 322 Factor a perfect square on the left side: (q + -12)(q + -12) = 322 Calculate the square root of the right side: 17.944358445 Break this problem into two subproblems by setting (q + -12) equal to 17.944358445 and -17.944358445.Subproblem 1
q + -12 = 17.944358445 Simplifying q + -12 = 17.944358445 Reorder the terms: -12 + q = 17.944358445 Solving -12 + q = 17.944358445 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 = 17.944358445 + 12 Combine like terms: -12 + 12 = 0 0 + q = 17.944358445 + 12 q = 17.944358445 + 12 Combine like terms: 17.944358445 + 12 = 29.944358445 q = 29.944358445 Simplifying q = 29.944358445Subproblem 2
q + -12 = -17.944358445 Simplifying q + -12 = -17.944358445 Reorder the terms: -12 + q = -17.944358445 Solving -12 + q = -17.944358445 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 = -17.944358445 + 12 Combine like terms: -12 + 12 = 0 0 + q = -17.944358445 + 12 q = -17.944358445 + 12 Combine like terms: -17.944358445 + 12 = -5.944358445 q = -5.944358445 Simplifying q = -5.944358445Solution
The solution to the problem is based on the solutions from the subproblems. q = {29.944358445, -5.944358445}
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