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Simplifying 8q2 + -4q2 = 9q + 24 Combine like terms: 8q2 + -4q2 = 4q2 4q2 = 9q + 24 Reorder the terms: 4q2 = 24 + 9q Solving 4q2 = 24 + 9q Solving for variable 'q'. Reorder the terms: -24 + -9q + 4q2 = 24 + 9q + -24 + -9q Reorder the terms: -24 + -9q + 4q2 = 24 + -24 + 9q + -9q Combine like terms: 24 + -24 = 0 -24 + -9q + 4q2 = 0 + 9q + -9q -24 + -9q + 4q2 = 9q + -9q Combine like terms: 9q + -9q = 0 -24 + -9q + 4q2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6 + -2.25q + q2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -2.25q + 6 + q2 = 0 + 6 Reorder the terms: -6 + 6 + -2.25q + q2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -2.25q + q2 = 0 + 6 -2.25q + q2 = 0 + 6 Combine like terms: 0 + 6 = 6 -2.25q + q2 = 6 The q term is -2.25q. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25q + 1.265625 + q2 = 6 + 1.265625 Reorder the terms: 1.265625 + -2.25q + q2 = 6 + 1.265625 Combine like terms: 6 + 1.265625 = 7.265625 1.265625 + -2.25q + q2 = 7.265625 Factor a perfect square on the left side: (q + -1.125)(q + -1.125) = 7.265625 Calculate the square root of the right side: 2.695482332 Break this problem into two subproblems by setting (q + -1.125) equal to 2.695482332 and -2.695482332.Subproblem 1
q + -1.125 = 2.695482332 Simplifying q + -1.125 = 2.695482332 Reorder the terms: -1.125 + q = 2.695482332 Solving -1.125 + q = 2.695482332 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + q = 2.695482332 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + q = 2.695482332 + 1.125 q = 2.695482332 + 1.125 Combine like terms: 2.695482332 + 1.125 = 3.820482332 q = 3.820482332 Simplifying q = 3.820482332Subproblem 2
q + -1.125 = -2.695482332 Simplifying q + -1.125 = -2.695482332 Reorder the terms: -1.125 + q = -2.695482332 Solving -1.125 + q = -2.695482332 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + q = -2.695482332 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + q = -2.695482332 + 1.125 q = -2.695482332 + 1.125 Combine like terms: -2.695482332 + 1.125 = -1.570482332 q = -1.570482332 Simplifying q = -1.570482332Solution
The solution to the problem is based on the solutions from the subproblems. q = {3.820482332, -1.570482332}
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