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Simplifying 2q2 + 37.5 = -20q Reorder the terms: 37.5 + 2q2 = -20q Solving 37.5 + 2q2 = -20q Solving for variable 'q'. Reorder the terms: 37.5 + 20q + 2q2 = -20q + 20q Combine like terms: -20q + 20q = 0 37.5 + 20q + 2q2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 18.75 + 10q + q2 = 0 Move the constant term to the right: Add '-18.75' to each side of the equation. 18.75 + 10q + -18.75 + q2 = 0 + -18.75 Reorder the terms: 18.75 + -18.75 + 10q + q2 = 0 + -18.75 Combine like terms: 18.75 + -18.75 = 0.00 0.00 + 10q + q2 = 0 + -18.75 10q + q2 = 0 + -18.75 Combine like terms: 0 + -18.75 = -18.75 10q + q2 = -18.75 The q term is 10q. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10q + 25 + q2 = -18.75 + 25 Reorder the terms: 25 + 10q + q2 = -18.75 + 25 Combine like terms: -18.75 + 25 = 6.25 25 + 10q + q2 = 6.25 Factor a perfect square on the left side: (q + 5)(q + 5) = 6.25 Calculate the square root of the right side: 2.5 Break this problem into two subproblems by setting (q + 5) equal to 2.5 and -2.5.Subproblem 1
q + 5 = 2.5 Simplifying q + 5 = 2.5 Reorder the terms: 5 + q = 2.5 Solving 5 + q = 2.5 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + q = 2.5 + -5 Combine like terms: 5 + -5 = 0 0 + q = 2.5 + -5 q = 2.5 + -5 Combine like terms: 2.5 + -5 = -2.5 q = -2.5 Simplifying q = -2.5Subproblem 2
q + 5 = -2.5 Simplifying q + 5 = -2.5 Reorder the terms: 5 + q = -2.5 Solving 5 + q = -2.5 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + q = -2.5 + -5 Combine like terms: 5 + -5 = 0 0 + q = -2.5 + -5 q = -2.5 + -5 Combine like terms: -2.5 + -5 = -7.5 q = -7.5 Simplifying q = -7.5Solution
The solution to the problem is based on the solutions from the subproblems. q = {-2.5, -7.5}
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