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Simplifying 574 = -16t2 + 64t + 512 Reorder the terms: 574 = 512 + 64t + -16t2 Solving 574 = 512 + 64t + -16t2 Solving for variable 't'. Combine like terms: 574 + -512 = 62 62 + -64t + 16t2 = 512 + 64t + -16t2 + -512 + -64t + 16t2 Reorder the terms: 62 + -64t + 16t2 = 512 + -512 + 64t + -64t + -16t2 + 16t2 Combine like terms: 512 + -512 = 0 62 + -64t + 16t2 = 0 + 64t + -64t + -16t2 + 16t2 62 + -64t + 16t2 = 64t + -64t + -16t2 + 16t2 Combine like terms: 64t + -64t = 0 62 + -64t + 16t2 = 0 + -16t2 + 16t2 62 + -64t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 62 + -64t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(31 + -32t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(31 + -32t + 8t2)' equal to zero and attempt to solve: Simplifying 31 + -32t + 8t2 = 0 Solving 31 + -32t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 3.875 + -4t + t2 = 0 Move the constant term to the right: Add '-3.875' to each side of the equation. 3.875 + -4t + -3.875 + t2 = 0 + -3.875 Reorder the terms: 3.875 + -3.875 + -4t + t2 = 0 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + -4t + t2 = 0 + -3.875 -4t + t2 = 0 + -3.875 Combine like terms: 0 + -3.875 = -3.875 -4t + t2 = -3.875 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -3.875 + 4 Reorder the terms: 4 + -4t + t2 = -3.875 + 4 Combine like terms: -3.875 + 4 = 0.125 4 + -4t + t2 = 0.125 Factor a perfect square on the left side: (t + -2)(t + -2) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (t + -2) equal to 0.353553391 and -0.353553391.Subproblem 1
t + -2 = 0.353553391 Simplifying t + -2 = 0.353553391 Reorder the terms: -2 + t = 0.353553391 Solving -2 + t = 0.353553391 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 0.353553391 + 2 Combine like terms: -2 + 2 = 0 0 + t = 0.353553391 + 2 t = 0.353553391 + 2 Combine like terms: 0.353553391 + 2 = 2.353553391 t = 2.353553391 Simplifying t = 2.353553391Subproblem 2
t + -2 = -0.353553391 Simplifying t + -2 = -0.353553391 Reorder the terms: -2 + t = -0.353553391 Solving -2 + t = -0.353553391 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -0.353553391 + 2 Combine like terms: -2 + 2 = 0 0 + t = -0.353553391 + 2 t = -0.353553391 + 2 Combine like terms: -0.353553391 + 2 = 1.646446609 t = 1.646446609 Simplifying t = 1.646446609Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.353553391, 1.646446609}Solution
t = {2.353553391, 1.646446609}
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