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Simplifying 0 = -16t2 + 64t + 384 Reorder the terms: 0 = 384 + 64t + -16t2 Solving 0 = 384 + 64t + -16t2 Solving for variable 't'. Combine like terms: 0 + -384 = -384 -384 + -64t + 16t2 = 384 + 64t + -16t2 + -384 + -64t + 16t2 Reorder the terms: -384 + -64t + 16t2 = 384 + -384 + 64t + -64t + -16t2 + 16t2 Combine like terms: 384 + -384 = 0 -384 + -64t + 16t2 = 0 + 64t + -64t + -16t2 + 16t2 -384 + -64t + 16t2 = 64t + -64t + -16t2 + 16t2 Combine like terms: 64t + -64t = 0 -384 + -64t + 16t2 = 0 + -16t2 + 16t2 -384 + -64t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -384 + -64t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(-24 + -4t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-24 + -4t + t2)' equal to zero and attempt to solve: Simplifying -24 + -4t + t2 = 0 Solving -24 + -4t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '24' to each side of the equation. -24 + -4t + 24 + t2 = 0 + 24 Reorder the terms: -24 + 24 + -4t + t2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + -4t + t2 = 0 + 24 -4t + t2 = 0 + 24 Combine like terms: 0 + 24 = 24 -4t + t2 = 24 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 = 24 + 4 Reorder the terms: 4 + -4t + t2 = 24 + 4 Combine like terms: 24 + 4 = 28 4 + -4t + t2 = 28 Factor a perfect square on the left side: (t + -2)(t + -2) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (t + -2) equal to 5.291502622 and -5.291502622.Subproblem 1
t + -2 = 5.291502622 Simplifying t + -2 = 5.291502622 Reorder the terms: -2 + t = 5.291502622 Solving -2 + t = 5.291502622 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 = 5.291502622 + 2 Combine like terms: -2 + 2 = 0 0 + t = 5.291502622 + 2 t = 5.291502622 + 2 Combine like terms: 5.291502622 + 2 = 7.291502622 t = 7.291502622 Simplifying t = 7.291502622Subproblem 2
t + -2 = -5.291502622 Simplifying t + -2 = -5.291502622 Reorder the terms: -2 + t = -5.291502622 Solving -2 + t = -5.291502622 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 = -5.291502622 + 2 Combine like terms: -2 + 2 = 0 0 + t = -5.291502622 + 2 t = -5.291502622 + 2 Combine like terms: -5.291502622 + 2 = -3.291502622 t = -3.291502622 Simplifying t = -3.291502622Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.291502622, -3.291502622}Solution
t = {7.291502622, -3.291502622}
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