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Simplifying 4q2 + 3q2 = 3q2 + -4q + 18 Combine like terms: 4q2 + 3q2 = 7q2 7q2 = 3q2 + -4q + 18 Reorder the terms: 7q2 = 18 + -4q + 3q2 Solving 7q2 = 18 + -4q + 3q2 Solving for variable 'q'. Reorder the terms: -18 + 4q + 7q2 + -3q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Combine like terms: 7q2 + -3q2 = 4q2 -18 + 4q + 4q2 = 18 + -4q + 3q2 + -18 + 4q + -3q2 Reorder the terms: -18 + 4q + 4q2 = 18 + -18 + -4q + 4q + 3q2 + -3q2 Combine like terms: 18 + -18 = 0 -18 + 4q + 4q2 = 0 + -4q + 4q + 3q2 + -3q2 -18 + 4q + 4q2 = -4q + 4q + 3q2 + -3q2 Combine like terms: -4q + 4q = 0 -18 + 4q + 4q2 = 0 + 3q2 + -3q2 -18 + 4q + 4q2 = 3q2 + -3q2 Combine like terms: 3q2 + -3q2 = 0 -18 + 4q + 4q2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-9 + 2q + 2q2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-9 + 2q + 2q2)' equal to zero and attempt to solve: Simplifying -9 + 2q + 2q2 = 0 Solving -9 + 2q + 2q2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + q + q2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + q + 4.5 + q2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + q + q2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + q + q2 = 0 + 4.5 q + q2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 q + q2 = 4.5 The q term is q. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + q2 = 4.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + q2 = 4.5 + 0.25 Combine like terms: 4.5 + 0.25 = 4.75 1.25 + q2 = 4.75 Factor a perfect square on the left side: (q + 0.5)(q + 0.5) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (q + 0.5) equal to 2.179449472 and -2.179449472.Subproblem 1
q + 0.5 = 2.179449472 Simplifying q + 0.5 = 2.179449472 Reorder the terms: 0.5 + q = 2.179449472 Solving 0.5 + q = 2.179449472 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + q = 2.179449472 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + q = 2.179449472 + -0.5 q = 2.179449472 + -0.5 Combine like terms: 2.179449472 + -0.5 = 1.679449472 q = 1.679449472 Simplifying q = 1.679449472Subproblem 2
q + 0.5 = -2.179449472 Simplifying q + 0.5 = -2.179449472 Reorder the terms: 0.5 + q = -2.179449472 Solving 0.5 + q = -2.179449472 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + q = -2.179449472 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + q = -2.179449472 + -0.5 q = -2.179449472 + -0.5 Combine like terms: -2.179449472 + -0.5 = -2.679449472 q = -2.679449472 Simplifying q = -2.679449472Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.679449472, -2.679449472}Solution
q = {1.679449472, -2.679449472}
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