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Simplifying -7 = 3(q + -5) * q Reorder the terms: -7 = 3(-5 + q) * q Reorder the terms for easier multiplication: -7 = 3q(-5 + q) -7 = (-5 * 3q + q * 3q) -7 = (-15q + 3q2) Solving -7 = -15q + 3q2 Solving for variable 'q'. Reorder the terms: -7 + 15q + -3q2 = -15q + 15q + 3q2 + -3q2 Combine like terms: -15q + 15q = 0 -7 + 15q + -3q2 = 0 + 3q2 + -3q2 -7 + 15q + -3q2 = 3q2 + -3q2 Combine like terms: 3q2 + -3q2 = 0 -7 + 15q + -3q2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 2.333333333 + -5q + q2 = 0 Move the constant term to the right: Add '-2.333333333' to each side of the equation. 2.333333333 + -5q + -2.333333333 + q2 = 0 + -2.333333333 Reorder the terms: 2.333333333 + -2.333333333 + -5q + q2 = 0 + -2.333333333 Combine like terms: 2.333333333 + -2.333333333 = 0.000000000 0.000000000 + -5q + q2 = 0 + -2.333333333 -5q + q2 = 0 + -2.333333333 Combine like terms: 0 + -2.333333333 = -2.333333333 -5q + q2 = -2.333333333 The q term is -5q. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5q + 6.25 + q2 = -2.333333333 + 6.25 Reorder the terms: 6.25 + -5q + q2 = -2.333333333 + 6.25 Combine like terms: -2.333333333 + 6.25 = 3.916666667 6.25 + -5q + q2 = 3.916666667 Factor a perfect square on the left side: (q + -2.5)(q + -2.5) = 3.916666667 Calculate the square root of the right side: 1.979057015 Break this problem into two subproblems by setting (q + -2.5) equal to 1.979057015 and -1.979057015.Subproblem 1
q + -2.5 = 1.979057015 Simplifying q + -2.5 = 1.979057015 Reorder the terms: -2.5 + q = 1.979057015 Solving -2.5 + q = 1.979057015 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + q = 1.979057015 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + q = 1.979057015 + 2.5 q = 1.979057015 + 2.5 Combine like terms: 1.979057015 + 2.5 = 4.479057015 q = 4.479057015 Simplifying q = 4.479057015Subproblem 2
q + -2.5 = -1.979057015 Simplifying q + -2.5 = -1.979057015 Reorder the terms: -2.5 + q = -1.979057015 Solving -2.5 + q = -1.979057015 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + q = -1.979057015 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + q = -1.979057015 + 2.5 q = -1.979057015 + 2.5 Combine like terms: -1.979057015 + 2.5 = 0.520942985 q = 0.520942985 Simplifying q = 0.520942985Solution
The solution to the problem is based on the solutions from the subproblems. q = {4.479057015, 0.520942985}
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