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Simplifying -0.012q2 + -8q = -410 Reorder the terms: -8q + -0.012q2 = -410 Solving -8q + -0.012q2 = -410 Solving for variable 'q'. Reorder the terms: 410 + -8q + -0.012q2 = -410 + 410 Combine like terms: -410 + 410 = 0 410 + -8q + -0.012q2 = 0 Begin completing the square. Divide all terms by -0.012 the coefficient of the squared term: Divide each side by '-0.012'. -34166.66667 + 666.6666667q + q2 = 0 Move the constant term to the right: Add '34166.66667' to each side of the equation. -34166.66667 + 666.6666667q + 34166.66667 + q2 = 0 + 34166.66667 Reorder the terms: -34166.66667 + 34166.66667 + 666.6666667q + q2 = 0 + 34166.66667 Combine like terms: -34166.66667 + 34166.66667 = 0.00000 0.00000 + 666.6666667q + q2 = 0 + 34166.66667 666.6666667q + q2 = 0 + 34166.66667 Combine like terms: 0 + 34166.66667 = 34166.66667 666.6666667q + q2 = 34166.66667 The q term is 666.6666667q. Take half its coefficient (333.3333334). Square it (111111.1112) and add it to both sides. Add '111111.1112' to each side of the equation. 666.6666667q + 111111.1112 + q2 = 34166.66667 + 111111.1112 Reorder the terms: 111111.1112 + 666.6666667q + q2 = 34166.66667 + 111111.1112 Combine like terms: 34166.66667 + 111111.1112 = 145277.77787 111111.1112 + 666.6666667q + q2 = 145277.77787 Factor a perfect square on the left side: (q + 333.3333334)(q + 333.3333334) = 145277.77787 Calculate the square root of the right side: 381.153220989 Break this problem into two subproblems by setting (q + 333.3333334) equal to 381.153220989 and -381.153220989.Subproblem 1
q + 333.3333334 = 381.153220989 Simplifying q + 333.3333334 = 381.153220989 Reorder the terms: 333.3333334 + q = 381.153220989 Solving 333.3333334 + q = 381.153220989 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-333.3333334' to each side of the equation. 333.3333334 + -333.3333334 + q = 381.153220989 + -333.3333334 Combine like terms: 333.3333334 + -333.3333334 = 0.0000000 0.0000000 + q = 381.153220989 + -333.3333334 q = 381.153220989 + -333.3333334 Combine like terms: 381.153220989 + -333.3333334 = 47.819887589 q = 47.819887589 Simplifying q = 47.819887589Subproblem 2
q + 333.3333334 = -381.153220989 Simplifying q + 333.3333334 = -381.153220989 Reorder the terms: 333.3333334 + q = -381.153220989 Solving 333.3333334 + q = -381.153220989 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-333.3333334' to each side of the equation. 333.3333334 + -333.3333334 + q = -381.153220989 + -333.3333334 Combine like terms: 333.3333334 + -333.3333334 = 0.0000000 0.0000000 + q = -381.153220989 + -333.3333334 q = -381.153220989 + -333.3333334 Combine like terms: -381.153220989 + -333.3333334 = -714.486554389 q = -714.486554389 Simplifying q = -714.486554389Solution
The solution to the problem is based on the solutions from the subproblems. q = {47.819887589, -714.486554389}
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