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Simplifying 3q(q + -5) = 2(q + 5) Reorder the terms: 3q(-5 + q) = 2(q + 5) (-5 * 3q + q * 3q) = 2(q + 5) (-15q + 3q2) = 2(q + 5) Reorder the terms: -15q + 3q2 = 2(5 + q) -15q + 3q2 = (5 * 2 + q * 2) -15q + 3q2 = (10 + 2q) Solving -15q + 3q2 = 10 + 2q Solving for variable 'q'. Reorder the terms: -10 + -15q + -2q + 3q2 = 10 + 2q + -10 + -2q Combine like terms: -15q + -2q = -17q -10 + -17q + 3q2 = 10 + 2q + -10 + -2q Reorder the terms: -10 + -17q + 3q2 = 10 + -10 + 2q + -2q Combine like terms: 10 + -10 = 0 -10 + -17q + 3q2 = 0 + 2q + -2q -10 + -17q + 3q2 = 2q + -2q Combine like terms: 2q + -2q = 0 -10 + -17q + 3q2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3.333333333 + -5.666666667q + q2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + -5.666666667q + 3.333333333 + q2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + -5.666666667q + q2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + -5.666666667q + q2 = 0 + 3.333333333 -5.666666667q + q2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 -5.666666667q + q2 = 3.333333333 The q term is -5.666666667q. Take half its coefficient (-2.833333334). Square it (8.027777782) and add it to both sides. Add '8.027777782' to each side of the equation. -5.666666667q + 8.027777782 + q2 = 3.333333333 + 8.027777782 Reorder the terms: 8.027777782 + -5.666666667q + q2 = 3.333333333 + 8.027777782 Combine like terms: 3.333333333 + 8.027777782 = 11.361111115 8.027777782 + -5.666666667q + q2 = 11.361111115 Factor a perfect square on the left side: (q + -2.833333334)(q + -2.833333334) = 11.361111115 Calculate the square root of the right side: 3.370624737 Break this problem into two subproblems by setting (q + -2.833333334) equal to 3.370624737 and -3.370624737.Subproblem 1
q + -2.833333334 = 3.370624737 Simplifying q + -2.833333334 = 3.370624737 Reorder the terms: -2.833333334 + q = 3.370624737 Solving -2.833333334 + q = 3.370624737 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.833333334' to each side of the equation. -2.833333334 + 2.833333334 + q = 3.370624737 + 2.833333334 Combine like terms: -2.833333334 + 2.833333334 = 0.000000000 0.000000000 + q = 3.370624737 + 2.833333334 q = 3.370624737 + 2.833333334 Combine like terms: 3.370624737 + 2.833333334 = 6.203958071 q = 6.203958071 Simplifying q = 6.203958071Subproblem 2
q + -2.833333334 = -3.370624737 Simplifying q + -2.833333334 = -3.370624737 Reorder the terms: -2.833333334 + q = -3.370624737 Solving -2.833333334 + q = -3.370624737 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.833333334' to each side of the equation. -2.833333334 + 2.833333334 + q = -3.370624737 + 2.833333334 Combine like terms: -2.833333334 + 2.833333334 = 0.000000000 0.000000000 + q = -3.370624737 + 2.833333334 q = -3.370624737 + 2.833333334 Combine like terms: -3.370624737 + 2.833333334 = -0.537291403 q = -0.537291403 Simplifying q = -0.537291403Solution
The solution to the problem is based on the solutions from the subproblems. q = {6.203958071, -0.537291403}
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