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Simplifying 16t2 + 75t + 80 = 0 Reorder the terms: 80 + 75t + 16t2 = 0 Solving 80 + 75t + 16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 5 + 4.6875t + t2 = 0 Move the constant term to the right: Add '-5' to each side of the equation. 5 + 4.6875t + -5 + t2 = 0 + -5 Reorder the terms: 5 + -5 + 4.6875t + t2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 4.6875t + t2 = 0 + -5 4.6875t + t2 = 0 + -5 Combine like terms: 0 + -5 = -5 4.6875t + t2 = -5 The t term is 4.6875t. Take half its coefficient (2.34375). Square it (5.493164063) and add it to both sides. Add '5.493164063' to each side of the equation. 4.6875t + 5.493164063 + t2 = -5 + 5.493164063 Reorder the terms: 5.493164063 + 4.6875t + t2 = -5 + 5.493164063 Combine like terms: -5 + 5.493164063 = 0.493164063 5.493164063 + 4.6875t + t2 = 0.493164063 Factor a perfect square on the left side: (t + 2.34375)(t + 2.34375) = 0.493164063 Calculate the square root of the right side: 0.702256408 Break this problem into two subproblems by setting (t + 2.34375) equal to 0.702256408 and -0.702256408.Subproblem 1
t + 2.34375 = 0.702256408 Simplifying t + 2.34375 = 0.702256408 Reorder the terms: 2.34375 + t = 0.702256408 Solving 2.34375 + t = 0.702256408 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.34375' to each side of the equation. 2.34375 + -2.34375 + t = 0.702256408 + -2.34375 Combine like terms: 2.34375 + -2.34375 = 0.00000 0.00000 + t = 0.702256408 + -2.34375 t = 0.702256408 + -2.34375 Combine like terms: 0.702256408 + -2.34375 = -1.641493592 t = -1.641493592 Simplifying t = -1.641493592Subproblem 2
t + 2.34375 = -0.702256408 Simplifying t + 2.34375 = -0.702256408 Reorder the terms: 2.34375 + t = -0.702256408 Solving 2.34375 + t = -0.702256408 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.34375' to each side of the equation. 2.34375 + -2.34375 + t = -0.702256408 + -2.34375 Combine like terms: 2.34375 + -2.34375 = 0.00000 0.00000 + t = -0.702256408 + -2.34375 t = -0.702256408 + -2.34375 Combine like terms: -0.702256408 + -2.34375 = -3.046006408 t = -3.046006408 Simplifying t = -3.046006408Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.641493592, -3.046006408}
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