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Simplifying 16t2 + -64t + -512 = 560 Reorder the terms: -512 + -64t + 16t2 = 560 Solving -512 + -64t + 16t2 = 560 Solving for variable 't'. Reorder the terms: -512 + -560 + -64t + 16t2 = 560 + -560 Combine like terms: -512 + -560 = -1072 -1072 + -64t + 16t2 = 560 + -560 Combine like terms: 560 + -560 = 0 -1072 + -64t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(-67 + -4t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-67 + -4t + t2)' equal to zero and attempt to solve: Simplifying -67 + -4t + t2 = 0 Solving -67 + -4t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '67' to each side of the equation. -67 + -4t + 67 + t2 = 0 + 67 Reorder the terms: -67 + 67 + -4t + t2 = 0 + 67 Combine like terms: -67 + 67 = 0 0 + -4t + t2 = 0 + 67 -4t + t2 = 0 + 67 Combine like terms: 0 + 67 = 67 -4t + t2 = 67 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 = 67 + 4 Reorder the terms: 4 + -4t + t2 = 67 + 4 Combine like terms: 67 + 4 = 71 4 + -4t + t2 = 71 Factor a perfect square on the left side: (t + -2)(t + -2) = 71 Calculate the square root of the right side: 8.426149773 Break this problem into two subproblems by setting (t + -2) equal to 8.426149773 and -8.426149773.Subproblem 1
t + -2 = 8.426149773 Simplifying t + -2 = 8.426149773 Reorder the terms: -2 + t = 8.426149773 Solving -2 + t = 8.426149773 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 = 8.426149773 + 2 Combine like terms: -2 + 2 = 0 0 + t = 8.426149773 + 2 t = 8.426149773 + 2 Combine like terms: 8.426149773 + 2 = 10.426149773 t = 10.426149773 Simplifying t = 10.426149773Subproblem 2
t + -2 = -8.426149773 Simplifying t + -2 = -8.426149773 Reorder the terms: -2 + t = -8.426149773 Solving -2 + t = -8.426149773 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 = -8.426149773 + 2 Combine like terms: -2 + 2 = 0 0 + t = -8.426149773 + 2 t = -8.426149773 + 2 Combine like terms: -8.426149773 + 2 = -6.426149773 t = -6.426149773 Simplifying t = -6.426149773Solution
The solution to the problem is based on the solutions from the subproblems. t = {10.426149773, -6.426149773}Solution
t = {10.426149773, -6.426149773}
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