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Simplifying 5q + -15 = q2 + 4q * 2q + -27 Reorder the terms: -15 + 5q = q2 + 4q * 2q + -27 Reorder the terms for easier multiplication: -15 + 5q = q2 + 4 * 2q * q + -27 Multiply 4 * 2 -15 + 5q = q2 + 8q * q + -27 Multiply q * q -15 + 5q = q2 + 8q2 + -27 Reorder the terms: -15 + 5q = -27 + q2 + 8q2 Combine like terms: q2 + 8q2 = 9q2 -15 + 5q = -27 + 9q2 Solving -15 + 5q = -27 + 9q2 Solving for variable 'q'. Reorder the terms: -15 + 27 + 5q + -9q2 = -27 + 9q2 + 27 + -9q2 Combine like terms: -15 + 27 = 12 12 + 5q + -9q2 = -27 + 9q2 + 27 + -9q2 Reorder the terms: 12 + 5q + -9q2 = -27 + 27 + 9q2 + -9q2 Combine like terms: -27 + 27 = 0 12 + 5q + -9q2 = 0 + 9q2 + -9q2 12 + 5q + -9q2 = 9q2 + -9q2 Combine like terms: 9q2 + -9q2 = 0 12 + 5q + -9q2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -1.333333333 + -0.5555555556q + q2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -0.5555555556q + 1.333333333 + q2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -0.5555555556q + q2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -0.5555555556q + q2 = 0 + 1.333333333 -0.5555555556q + q2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -0.5555555556q + q2 = 1.333333333 The q term is -0.5555555556q. Take half its coefficient (-0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. -0.5555555556q + 0.07716049384 + q2 = 1.333333333 + 0.07716049384 Reorder the terms: 0.07716049384 + -0.5555555556q + q2 = 1.333333333 + 0.07716049384 Combine like terms: 1.333333333 + 0.07716049384 = 1.41049382684 0.07716049384 + -0.5555555556q + q2 = 1.41049382684 Factor a perfect square on the left side: (q + -0.2777777778)(q + -0.2777777778) = 1.41049382684 Calculate the square root of the right side: 1.187642129 Break this problem into two subproblems by setting (q + -0.2777777778) equal to 1.187642129 and -1.187642129.Subproblem 1
q + -0.2777777778 = 1.187642129 Simplifying q + -0.2777777778 = 1.187642129 Reorder the terms: -0.2777777778 + q = 1.187642129 Solving -0.2777777778 + q = 1.187642129 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + q = 1.187642129 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + q = 1.187642129 + 0.2777777778 q = 1.187642129 + 0.2777777778 Combine like terms: 1.187642129 + 0.2777777778 = 1.4654199068 q = 1.4654199068 Simplifying q = 1.4654199068Subproblem 2
q + -0.2777777778 = -1.187642129 Simplifying q + -0.2777777778 = -1.187642129 Reorder the terms: -0.2777777778 + q = -1.187642129 Solving -0.2777777778 + q = -1.187642129 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + q = -1.187642129 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + q = -1.187642129 + 0.2777777778 q = -1.187642129 + 0.2777777778 Combine like terms: -1.187642129 + 0.2777777778 = -0.9098643512 q = -0.9098643512 Simplifying q = -0.9098643512Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.4654199068, -0.9098643512}
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