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Simplifying -16t2 + 64t + 8 = t Reorder the terms: 8 + 64t + -16t2 = t Solving 8 + 64t + -16t2 = t Solving for variable 't'. Reorder the terms: 8 + 64t + -1t + -16t2 = t + -1t Combine like terms: 64t + -1t = 63t 8 + 63t + -16t2 = t + -1t Combine like terms: t + -1t = 0 8 + 63t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.5 + -3.9375t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -3.9375t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -3.9375t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -3.9375t + t2 = 0 + 0.5 -3.9375t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -3.9375t + t2 = 0.5 The t term is -3.9375t. Take half its coefficient (-1.96875). Square it (3.875976563) and add it to both sides. Add '3.875976563' to each side of the equation. -3.9375t + 3.875976563 + t2 = 0.5 + 3.875976563 Reorder the terms: 3.875976563 + -3.9375t + t2 = 0.5 + 3.875976563 Combine like terms: 0.5 + 3.875976563 = 4.375976563 3.875976563 + -3.9375t + t2 = 4.375976563 Factor a perfect square on the left side: (t + -1.96875)(t + -1.96875) = 4.375976563 Calculate the square root of the right side: 2.091883497 Break this problem into two subproblems by setting (t + -1.96875) equal to 2.091883497 and -2.091883497.Subproblem 1
t + -1.96875 = 2.091883497 Simplifying t + -1.96875 = 2.091883497 Reorder the terms: -1.96875 + t = 2.091883497 Solving -1.96875 + t = 2.091883497 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.96875' to each side of the equation. -1.96875 + 1.96875 + t = 2.091883497 + 1.96875 Combine like terms: -1.96875 + 1.96875 = 0.00000 0.00000 + t = 2.091883497 + 1.96875 t = 2.091883497 + 1.96875 Combine like terms: 2.091883497 + 1.96875 = 4.060633497 t = 4.060633497 Simplifying t = 4.060633497Subproblem 2
t + -1.96875 = -2.091883497 Simplifying t + -1.96875 = -2.091883497 Reorder the terms: -1.96875 + t = -2.091883497 Solving -1.96875 + t = -2.091883497 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.96875' to each side of the equation. -1.96875 + 1.96875 + t = -2.091883497 + 1.96875 Combine like terms: -1.96875 + 1.96875 = 0.00000 0.00000 + t = -2.091883497 + 1.96875 t = -2.091883497 + 1.96875 Combine like terms: -2.091883497 + 1.96875 = -0.123133497 t = -0.123133497 Simplifying t = -0.123133497Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.060633497, -0.123133497}
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