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Simplifying 7q + 3 = 5q + 2q(q + -4) Reorder the terms: 3 + 7q = 5q + 2q(q + -4) Reorder the terms: 3 + 7q = 5q + 2q(-4 + q) 3 + 7q = 5q + (-4 * 2q + q * 2q) 3 + 7q = 5q + (-8q + 2q2) Combine like terms: 5q + -8q = -3q 3 + 7q = -3q + 2q2 Solving 3 + 7q = -3q + 2q2 Solving for variable 'q'. Combine like terms: 7q + 3q = 10q 3 + 10q + -2q2 = -3q + 2q2 + 3q + -2q2 Reorder the terms: 3 + 10q + -2q2 = -3q + 3q + 2q2 + -2q2 Combine like terms: -3q + 3q = 0 3 + 10q + -2q2 = 0 + 2q2 + -2q2 3 + 10q + -2q2 = 2q2 + -2q2 Combine like terms: 2q2 + -2q2 = 0 3 + 10q + -2q2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1.5 + -5q + q2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -5q + 1.5 + q2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -5q + q2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -5q + q2 = 0 + 1.5 -5q + q2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -5q + q2 = 1.5 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 = 1.5 + 6.25 Reorder the terms: 6.25 + -5q + q2 = 1.5 + 6.25 Combine like terms: 1.5 + 6.25 = 7.75 6.25 + -5q + q2 = 7.75 Factor a perfect square on the left side: (q + -2.5)(q + -2.5) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (q + -2.5) equal to 2.783882181 and -2.783882181.Subproblem 1
q + -2.5 = 2.783882181 Simplifying q + -2.5 = 2.783882181 Reorder the terms: -2.5 + q = 2.783882181 Solving -2.5 + q = 2.783882181 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 = 2.783882181 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + q = 2.783882181 + 2.5 q = 2.783882181 + 2.5 Combine like terms: 2.783882181 + 2.5 = 5.283882181 q = 5.283882181 Simplifying q = 5.283882181Subproblem 2
q + -2.5 = -2.783882181 Simplifying q + -2.5 = -2.783882181 Reorder the terms: -2.5 + q = -2.783882181 Solving -2.5 + q = -2.783882181 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 = -2.783882181 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + q = -2.783882181 + 2.5 q = -2.783882181 + 2.5 Combine like terms: -2.783882181 + 2.5 = -0.283882181 q = -0.283882181 Simplifying q = -0.283882181Solution
The solution to the problem is based on the solutions from the subproblems. q = {5.283882181, -0.283882181}
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