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Simplifying -16t2 + 4096 = t Reorder the terms: 4096 + -16t2 = t Solving 4096 + -16t2 = t Solving for variable 't'. Reorder the terms: 4096 + -1t + -16t2 = t + -1t Combine like terms: t + -1t = 0 4096 + -1t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -256 + 0.0625t + t2 = 0 Move the constant term to the right: Add '256' to each side of the equation. -256 + 0.0625t + 256 + t2 = 0 + 256 Reorder the terms: -256 + 256 + 0.0625t + t2 = 0 + 256 Combine like terms: -256 + 256 = 0 0 + 0.0625t + t2 = 0 + 256 0.0625t + t2 = 0 + 256 Combine like terms: 0 + 256 = 256 0.0625t + t2 = 256 The t term is 0.0625t. Take half its coefficient (0.03125). Square it (0.0009765625) and add it to both sides. Add '0.0009765625' to each side of the equation. 0.0625t + 0.0009765625 + t2 = 256 + 0.0009765625 Reorder the terms: 0.0009765625 + 0.0625t + t2 = 256 + 0.0009765625 Combine like terms: 256 + 0.0009765625 = 256.0009765625 0.0009765625 + 0.0625t + t2 = 256.0009765625 Factor a perfect square on the left side: (t + 0.03125)(t + 0.03125) = 256.0009765625 Calculate the square root of the right side: 16.000030518 Break this problem into two subproblems by setting (t + 0.03125) equal to 16.000030518 and -16.000030518.Subproblem 1
t + 0.03125 = 16.000030518 Simplifying t + 0.03125 = 16.000030518 Reorder the terms: 0.03125 + t = 16.000030518 Solving 0.03125 + t = 16.000030518 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.03125' to each side of the equation. 0.03125 + -0.03125 + t = 16.000030518 + -0.03125 Combine like terms: 0.03125 + -0.03125 = 0.00000 0.00000 + t = 16.000030518 + -0.03125 t = 16.000030518 + -0.03125 Combine like terms: 16.000030518 + -0.03125 = 15.968780518 t = 15.968780518 Simplifying t = 15.968780518Subproblem 2
t + 0.03125 = -16.000030518 Simplifying t + 0.03125 = -16.000030518 Reorder the terms: 0.03125 + t = -16.000030518 Solving 0.03125 + t = -16.000030518 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.03125' to each side of the equation. 0.03125 + -0.03125 + t = -16.000030518 + -0.03125 Combine like terms: 0.03125 + -0.03125 = 0.00000 0.00000 + t = -16.000030518 + -0.03125 t = -16.000030518 + -0.03125 Combine like terms: -16.000030518 + -0.03125 = -16.031280518 t = -16.031280518 Simplifying t = -16.031280518Solution
The solution to the problem is based on the solutions from the subproblems. t = {15.968780518, -16.031280518}
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