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Simplifying 0.5q2 + 3.5 = -4q Reorder the terms: 3.5 + 0.5q2 = -4q Solving 3.5 + 0.5q2 = -4q Solving for variable 'q'. Reorder the terms: 3.5 + 4q + 0.5q2 = -4q + 4q Combine like terms: -4q + 4q = 0 3.5 + 4q + 0.5q2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 7 + 8q + q2 = 0 Move the constant term to the right: Add '-7' to each side of the equation. 7 + 8q + -7 + q2 = 0 + -7 Reorder the terms: 7 + -7 + 8q + q2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + 8q + q2 = 0 + -7 8q + q2 = 0 + -7 Combine like terms: 0 + -7 = -7 8q + q2 = -7 The q term is 8q. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8q + 16 + q2 = -7 + 16 Reorder the terms: 16 + 8q + q2 = -7 + 16 Combine like terms: -7 + 16 = 9 16 + 8q + q2 = 9 Factor a perfect square on the left side: (q + 4)(q + 4) = 9 Calculate the square root of the right side: 3 Break this problem into two subproblems by setting (q + 4) equal to 3 and -3.Subproblem 1
q + 4 = 3 Simplifying q + 4 = 3 Reorder the terms: 4 + q = 3 Solving 4 + q = 3 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + q = 3 + -4 Combine like terms: 4 + -4 = 0 0 + q = 3 + -4 q = 3 + -4 Combine like terms: 3 + -4 = -1 q = -1 Simplifying q = -1Subproblem 2
q + 4 = -3 Simplifying q + 4 = -3 Reorder the terms: 4 + q = -3 Solving 4 + q = -3 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + q = -3 + -4 Combine like terms: 4 + -4 = 0 0 + q = -3 + -4 q = -3 + -4 Combine like terms: -3 + -4 = -7 q = -7 Simplifying q = -7Solution
The solution to the problem is based on the solutions from the subproblems. q = {-1, -7}
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