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Simplifying 0 = -16t2 + 40t + -22 Reorder the terms: 0 = -22 + 40t + -16t2 Solving 0 = -22 + 40t + -16t2 Solving for variable 't'. Combine like terms: 0 + 22 = 22 22 + -40t + 16t2 = -22 + 40t + -16t2 + 22 + -40t + 16t2 Reorder the terms: 22 + -40t + 16t2 = -22 + 22 + 40t + -40t + -16t2 + 16t2 Combine like terms: -22 + 22 = 0 22 + -40t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 22 + -40t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 22 + -40t + 16t2 = 0 + -16t2 + 16t2 22 + -40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 22 + -40t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(11 + -20t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(11 + -20t + 8t2)' equal to zero and attempt to solve: Simplifying 11 + -20t + 8t2 = 0 Solving 11 + -20t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.375 + -2.5t + t2 = 0 Move the constant term to the right: Add '-1.375' to each side of the equation. 1.375 + -2.5t + -1.375 + t2 = 0 + -1.375 Reorder the terms: 1.375 + -1.375 + -2.5t + t2 = 0 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + -2.5t + t2 = 0 + -1.375 -2.5t + t2 = 0 + -1.375 Combine like terms: 0 + -1.375 = -1.375 -2.5t + t2 = -1.375 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = -1.375 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = -1.375 + 1.5625 Combine like terms: -1.375 + 1.5625 = 0.1875 1.5625 + -2.5t + t2 = 0.1875 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (t + -1.25) equal to 0.433012702 and -0.433012702.Subproblem 1
t + -1.25 = 0.433012702 Simplifying t + -1.25 = 0.433012702 Reorder the terms: -1.25 + t = 0.433012702 Solving -1.25 + t = 0.433012702 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 0.433012702 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 0.433012702 + 1.25 t = 0.433012702 + 1.25 Combine like terms: 0.433012702 + 1.25 = 1.683012702 t = 1.683012702 Simplifying t = 1.683012702Subproblem 2
t + -1.25 = -0.433012702 Simplifying t + -1.25 = -0.433012702 Reorder the terms: -1.25 + t = -0.433012702 Solving -1.25 + t = -0.433012702 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -0.433012702 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -0.433012702 + 1.25 t = -0.433012702 + 1.25 Combine like terms: -0.433012702 + 1.25 = 0.816987298 t = 0.816987298 Simplifying t = 0.816987298Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.683012702, 0.816987298}Solution
t = {1.683012702, 0.816987298}
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