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Simplifying 2q2 + 4q + -1 = 7q2 + -7 + 1 Reorder the terms: -1 + 4q + 2q2 = 7q2 + -7 + 1 Reorder the terms: -1 + 4q + 2q2 = -7 + 1 + 7q2 Combine like terms: -7 + 1 = -6 -1 + 4q + 2q2 = -6 + 7q2 Solving -1 + 4q + 2q2 = -6 + 7q2 Solving for variable 'q'. Reorder the terms: -1 + 6 + 4q + 2q2 + -7q2 = -6 + 7q2 + 6 + -7q2 Combine like terms: -1 + 6 = 5 5 + 4q + 2q2 + -7q2 = -6 + 7q2 + 6 + -7q2 Combine like terms: 2q2 + -7q2 = -5q2 5 + 4q + -5q2 = -6 + 7q2 + 6 + -7q2 Reorder the terms: 5 + 4q + -5q2 = -6 + 6 + 7q2 + -7q2 Combine like terms: -6 + 6 = 0 5 + 4q + -5q2 = 0 + 7q2 + -7q2 5 + 4q + -5q2 = 7q2 + -7q2 Combine like terms: 7q2 + -7q2 = 0 5 + 4q + -5q2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -1 + -0.8q + q2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.8q + 1 + q2 = 0 + 1 Reorder the terms: -1 + 1 + -0.8q + q2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.8q + q2 = 0 + 1 -0.8q + q2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.8q + q2 = 1 The q term is -0.8q. Take half its coefficient (-0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. -0.8q + 0.16 + q2 = 1 + 0.16 Reorder the terms: 0.16 + -0.8q + q2 = 1 + 0.16 Combine like terms: 1 + 0.16 = 1.16 0.16 + -0.8q + q2 = 1.16 Factor a perfect square on the left side: (q + -0.4)(q + -0.4) = 1.16 Calculate the square root of the right side: 1.077032961 Break this problem into two subproblems by setting (q + -0.4) equal to 1.077032961 and -1.077032961.Subproblem 1
q + -0.4 = 1.077032961 Simplifying q + -0.4 = 1.077032961 Reorder the terms: -0.4 + q = 1.077032961 Solving -0.4 + q = 1.077032961 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + q = 1.077032961 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + q = 1.077032961 + 0.4 q = 1.077032961 + 0.4 Combine like terms: 1.077032961 + 0.4 = 1.477032961 q = 1.477032961 Simplifying q = 1.477032961Subproblem 2
q + -0.4 = -1.077032961 Simplifying q + -0.4 = -1.077032961 Reorder the terms: -0.4 + q = -1.077032961 Solving -0.4 + q = -1.077032961 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + q = -1.077032961 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + q = -1.077032961 + 0.4 q = -1.077032961 + 0.4 Combine like terms: -1.077032961 + 0.4 = -0.677032961 q = -0.677032961 Simplifying q = -0.677032961Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.477032961, -0.677032961}
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