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Simplifying 2q(8q + -5) = 7 + 4q Reorder the terms: 2q(-5 + 8q) = 7 + 4q (-5 * 2q + 8q * 2q) = 7 + 4q (-10q + 16q2) = 7 + 4q Solving -10q + 16q2 = 7 + 4q Solving for variable 'q'. Reorder the terms: -7 + -10q + -4q + 16q2 = 7 + 4q + -7 + -4q Combine like terms: -10q + -4q = -14q -7 + -14q + 16q2 = 7 + 4q + -7 + -4q Reorder the terms: -7 + -14q + 16q2 = 7 + -7 + 4q + -4q Combine like terms: 7 + -7 = 0 -7 + -14q + 16q2 = 0 + 4q + -4q -7 + -14q + 16q2 = 4q + -4q Combine like terms: 4q + -4q = 0 -7 + -14q + 16q2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.4375 + -0.875q + q2 = 0 Move the constant term to the right: Add '0.4375' to each side of the equation. -0.4375 + -0.875q + 0.4375 + q2 = 0 + 0.4375 Reorder the terms: -0.4375 + 0.4375 + -0.875q + q2 = 0 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + -0.875q + q2 = 0 + 0.4375 -0.875q + q2 = 0 + 0.4375 Combine like terms: 0 + 0.4375 = 0.4375 -0.875q + q2 = 0.4375 The q term is -0.875q. Take half its coefficient (-0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. -0.875q + 0.19140625 + q2 = 0.4375 + 0.19140625 Reorder the terms: 0.19140625 + -0.875q + q2 = 0.4375 + 0.19140625 Combine like terms: 0.4375 + 0.19140625 = 0.62890625 0.19140625 + -0.875q + q2 = 0.62890625 Factor a perfect square on the left side: (q + -0.4375)(q + -0.4375) = 0.62890625 Calculate the square root of the right side: 0.793036096 Break this problem into two subproblems by setting (q + -0.4375) equal to 0.793036096 and -0.793036096.Subproblem 1
q + -0.4375 = 0.793036096 Simplifying q + -0.4375 = 0.793036096 Reorder the terms: -0.4375 + q = 0.793036096 Solving -0.4375 + q = 0.793036096 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + q = 0.793036096 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + q = 0.793036096 + 0.4375 q = 0.793036096 + 0.4375 Combine like terms: 0.793036096 + 0.4375 = 1.230536096 q = 1.230536096 Simplifying q = 1.230536096Subproblem 2
q + -0.4375 = -0.793036096 Simplifying q + -0.4375 = -0.793036096 Reorder the terms: -0.4375 + q = -0.793036096 Solving -0.4375 + q = -0.793036096 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + q = -0.793036096 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + q = -0.793036096 + 0.4375 q = -0.793036096 + 0.4375 Combine like terms: -0.793036096 + 0.4375 = -0.355536096 q = -0.355536096 Simplifying q = -0.355536096Solution
The solution to the problem is based on the solutions from the subproblems. q = {1.230536096, -0.355536096}
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