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Simplifying -16t2 + 14t + 240 = 0 Reorder the terms: 240 + 14t + -16t2 = 0 Solving 240 + 14t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(120 + 7t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(120 + 7t + -8t2)' equal to zero and attempt to solve: Simplifying 120 + 7t + -8t2 = 0 Solving 120 + 7t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -15 + -0.875t + t2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + -0.875t + 15 + t2 = 0 + 15 Reorder the terms: -15 + 15 + -0.875t + t2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + -0.875t + t2 = 0 + 15 -0.875t + t2 = 0 + 15 Combine like terms: 0 + 15 = 15 -0.875t + t2 = 15 The t term is -0.875t. Take half its coefficient (-0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. -0.875t + 0.19140625 + t2 = 15 + 0.19140625 Reorder the terms: 0.19140625 + -0.875t + t2 = 15 + 0.19140625 Combine like terms: 15 + 0.19140625 = 15.19140625 0.19140625 + -0.875t + t2 = 15.19140625 Factor a perfect square on the left side: (t + -0.4375)(t + -0.4375) = 15.19140625 Calculate the square root of the right side: 3.897615457 Break this problem into two subproblems by setting (t + -0.4375) equal to 3.897615457 and -3.897615457.Subproblem 1
t + -0.4375 = 3.897615457 Simplifying t + -0.4375 = 3.897615457 Reorder the terms: -0.4375 + t = 3.897615457 Solving -0.4375 + t = 3.897615457 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + t = 3.897615457 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + t = 3.897615457 + 0.4375 t = 3.897615457 + 0.4375 Combine like terms: 3.897615457 + 0.4375 = 4.335115457 t = 4.335115457 Simplifying t = 4.335115457Subproblem 2
t + -0.4375 = -3.897615457 Simplifying t + -0.4375 = -3.897615457 Reorder the terms: -0.4375 + t = -3.897615457 Solving -0.4375 + t = -3.897615457 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + t = -3.897615457 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + t = -3.897615457 + 0.4375 t = -3.897615457 + 0.4375 Combine like terms: -3.897615457 + 0.4375 = -3.460115457 t = -3.460115457 Simplifying t = -3.460115457Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.335115457, -3.460115457}Solution
t = {4.335115457, -3.460115457}
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