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Simplifying 384 = 16t2 + 64t Reorder the terms: 384 = 64t + 16t2 Solving 384 = 64t + 16t2 Solving for variable 't'. Reorder the terms: 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 + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(24 + -4t + -1t2)' equal to zero and attempt to solve: Simplifying 24 + -4t + -1t2 = 0 Solving 24 + -4t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -24 + 4t + t2 = 0 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 = 3.291502622 t = 3.291502622 Simplifying t = 3.291502622Subproblem 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 = -7.291502622 t = -7.291502622 Simplifying t = -7.291502622Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.291502622, -7.291502622}Solution
t = {3.291502622, -7.291502622}
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