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Simplifying 3 = -16t2 + 16t + 192 Reorder the terms: 3 = 192 + 16t + -16t2 Solving 3 = 192 + 16t + -16t2 Solving for variable 't'. Combine like terms: 3 + -192 = -189 -189 + -16t + 16t2 = 192 + 16t + -16t2 + -192 + -16t + 16t2 Reorder the terms: -189 + -16t + 16t2 = 192 + -192 + 16t + -16t + -16t2 + 16t2 Combine like terms: 192 + -192 = 0 -189 + -16t + 16t2 = 0 + 16t + -16t + -16t2 + 16t2 -189 + -16t + 16t2 = 16t + -16t + -16t2 + 16t2 Combine like terms: 16t + -16t = 0 -189 + -16t + 16t2 = 0 + -16t2 + 16t2 -189 + -16t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -189 + -16t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -11.8125 + -1t + t2 = 0 Move the constant term to the right: Add '11.8125' to each side of the equation. -11.8125 + -1t + 11.8125 + t2 = 0 + 11.8125 Reorder the terms: -11.8125 + 11.8125 + -1t + t2 = 0 + 11.8125 Combine like terms: -11.8125 + 11.8125 = 0.0000 0.0000 + -1t + t2 = 0 + 11.8125 -1t + t2 = 0 + 11.8125 Combine like terms: 0 + 11.8125 = 11.8125 -1t + t2 = 11.8125 The t term is -1t. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1t + 0.25 + t2 = 11.8125 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 11.8125 + 0.25 Combine like terms: 11.8125 + 0.25 = 12.0625 0.25 + -1t + t2 = 12.0625 Factor a perfect square on the left side: (t + -0.5)(t + -0.5) = 12.0625 Calculate the square root of the right side: 3.473110997 Break this problem into two subproblems by setting (t + -0.5) equal to 3.473110997 and -3.473110997.Subproblem 1
t + -0.5 = 3.473110997 Simplifying t + -0.5 = 3.473110997 Reorder the terms: -0.5 + t = 3.473110997 Solving -0.5 + t = 3.473110997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = 3.473110997 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 3.473110997 + 0.5 t = 3.473110997 + 0.5 Combine like terms: 3.473110997 + 0.5 = 3.973110997 t = 3.973110997 Simplifying t = 3.973110997Subproblem 2
t + -0.5 = -3.473110997 Simplifying t + -0.5 = -3.473110997 Reorder the terms: -0.5 + t = -3.473110997 Solving -0.5 + t = -3.473110997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = -3.473110997 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -3.473110997 + 0.5 t = -3.473110997 + 0.5 Combine like terms: -3.473110997 + 0.5 = -2.973110997 t = -2.973110997 Simplifying t = -2.973110997Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.973110997, -2.973110997}
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