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Simplifying 2(2t + -4) * t = 16 + -1t Reorder the terms: 2(-4 + 2t) * t = 16 + -1t Reorder the terms for easier multiplication: 2t(-4 + 2t) = 16 + -1t (-4 * 2t + 2t * 2t) = 16 + -1t (-8t + 4t2) = 16 + -1t Solving -8t + 4t2 = 16 + -1t Solving for variable 't'. Reorder the terms: -16 + -8t + t + 4t2 = 16 + -1t + -16 + t Combine like terms: -8t + t = -7t -16 + -7t + 4t2 = 16 + -1t + -16 + t Reorder the terms: -16 + -7t + 4t2 = 16 + -16 + -1t + t Combine like terms: 16 + -16 = 0 -16 + -7t + 4t2 = 0 + -1t + t -16 + -7t + 4t2 = -1t + t Combine like terms: -1t + t = 0 -16 + -7t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4 + -1.75t + t2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -1.75t + 4 + t2 = 0 + 4 Reorder the terms: -4 + 4 + -1.75t + t2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -1.75t + t2 = 0 + 4 -1.75t + t2 = 0 + 4 Combine like terms: 0 + 4 = 4 -1.75t + t2 = 4 The t term is -1.75t. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75t + 0.765625 + t2 = 4 + 0.765625 Reorder the terms: 0.765625 + -1.75t + t2 = 4 + 0.765625 Combine like terms: 4 + 0.765625 = 4.765625 0.765625 + -1.75t + t2 = 4.765625 Factor a perfect square on the left side: (t + -0.875)(t + -0.875) = 4.765625 Calculate the square root of the right side: 2.18303115 Break this problem into two subproblems by setting (t + -0.875) equal to 2.18303115 and -2.18303115.Subproblem 1
t + -0.875 = 2.18303115 Simplifying t + -0.875 = 2.18303115 Reorder the terms: -0.875 + t = 2.18303115 Solving -0.875 + t = 2.18303115 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = 2.18303115 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = 2.18303115 + 0.875 t = 2.18303115 + 0.875 Combine like terms: 2.18303115 + 0.875 = 3.05803115 t = 3.05803115 Simplifying t = 3.05803115Subproblem 2
t + -0.875 = -2.18303115 Simplifying t + -0.875 = -2.18303115 Reorder the terms: -0.875 + t = -2.18303115 Solving -0.875 + t = -2.18303115 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + t = -2.18303115 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + t = -2.18303115 + 0.875 t = -2.18303115 + 0.875 Combine like terms: -2.18303115 + 0.875 = -1.30803115 t = -1.30803115 Simplifying t = -1.30803115Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.05803115, -1.30803115}
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