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Simplifying 0 = 16t2 + -38t + -112 Reorder the terms: 0 = -112 + -38t + 16t2 Solving 0 = -112 + -38t + 16t2 Solving for variable 't'. Combine like terms: 0 + 112 = 112 112 + 38t + -16t2 = -112 + -38t + 16t2 + 112 + 38t + -16t2 Reorder the terms: 112 + 38t + -16t2 = -112 + 112 + -38t + 38t + 16t2 + -16t2 Combine like terms: -112 + 112 = 0 112 + 38t + -16t2 = 0 + -38t + 38t + 16t2 + -16t2 112 + 38t + -16t2 = -38t + 38t + 16t2 + -16t2 Combine like terms: -38t + 38t = 0 112 + 38t + -16t2 = 0 + 16t2 + -16t2 112 + 38t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 112 + 38t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(56 + 19t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(56 + 19t + -8t2)' equal to zero and attempt to solve: Simplifying 56 + 19t + -8t2 = 0 Solving 56 + 19t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -7 + -2.375t + t2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + -2.375t + 7 + t2 = 0 + 7 Reorder the terms: -7 + 7 + -2.375t + t2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -2.375t + t2 = 0 + 7 -2.375t + t2 = 0 + 7 Combine like terms: 0 + 7 = 7 -2.375t + t2 = 7 The t term is -2.375t. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375t + 1.41015625 + t2 = 7 + 1.41015625 Reorder the terms: 1.41015625 + -2.375t + t2 = 7 + 1.41015625 Combine like terms: 7 + 1.41015625 = 8.41015625 1.41015625 + -2.375t + t2 = 8.41015625 Factor a perfect square on the left side: (t + -1.1875)(t + -1.1875) = 8.41015625 Calculate the square root of the right side: 2.90002694 Break this problem into two subproblems by setting (t + -1.1875) equal to 2.90002694 and -2.90002694.Subproblem 1
t + -1.1875 = 2.90002694 Simplifying t + -1.1875 = 2.90002694 Reorder the terms: -1.1875 + t = 2.90002694 Solving -1.1875 + t = 2.90002694 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + t = 2.90002694 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = 2.90002694 + 1.1875 t = 2.90002694 + 1.1875 Combine like terms: 2.90002694 + 1.1875 = 4.08752694 t = 4.08752694 Simplifying t = 4.08752694Subproblem 2
t + -1.1875 = -2.90002694 Simplifying t + -1.1875 = -2.90002694 Reorder the terms: -1.1875 + t = -2.90002694 Solving -1.1875 + t = -2.90002694 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + t = -2.90002694 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = -2.90002694 + 1.1875 t = -2.90002694 + 1.1875 Combine like terms: -2.90002694 + 1.1875 = -1.71252694 t = -1.71252694 Simplifying t = -1.71252694Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.08752694, -1.71252694}Solution
t = {4.08752694, -1.71252694}
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