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Simplifying 38 = 16t2 + 48t + 2 Reorder the terms: 38 = 2 + 48t + 16t2 Solving 38 = 2 + 48t + 16t2 Solving for variable 't'. Combine like terms: 38 + -2 = 36 36 + -48t + -16t2 = 2 + 48t + 16t2 + -2 + -48t + -16t2 Reorder the terms: 36 + -48t + -16t2 = 2 + -2 + 48t + -48t + 16t2 + -16t2 Combine like terms: 2 + -2 = 0 36 + -48t + -16t2 = 0 + 48t + -48t + 16t2 + -16t2 36 + -48t + -16t2 = 48t + -48t + 16t2 + -16t2 Combine like terms: 48t + -48t = 0 36 + -48t + -16t2 = 0 + 16t2 + -16t2 36 + -48t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 36 + -48t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(9 + -12t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(9 + -12t + -4t2)' equal to zero and attempt to solve: Simplifying 9 + -12t + -4t2 = 0 Solving 9 + -12t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.25 + 3t + t2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + 3t + 2.25 + t2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + 3t + t2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + 3t + t2 = 0 + 2.25 3t + t2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 3t + t2 = 2.25 The t term is 3t. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3t + 2.25 + t2 = 2.25 + 2.25 Reorder the terms: 2.25 + 3t + t2 = 2.25 + 2.25 Combine like terms: 2.25 + 2.25 = 4.5 2.25 + 3t + t2 = 4.5 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 4.5 Calculate the square root of the right side: 2.121320344 Break this problem into two subproblems by setting (t + 1.5) equal to 2.121320344 and -2.121320344.Subproblem 1
t + 1.5 = 2.121320344 Simplifying t + 1.5 = 2.121320344 Reorder the terms: 1.5 + t = 2.121320344 Solving 1.5 + t = 2.121320344 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = 2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 2.121320344 + -1.5 t = 2.121320344 + -1.5 Combine like terms: 2.121320344 + -1.5 = 0.621320344 t = 0.621320344 Simplifying t = 0.621320344Subproblem 2
t + 1.5 = -2.121320344 Simplifying t + 1.5 = -2.121320344 Reorder the terms: 1.5 + t = -2.121320344 Solving 1.5 + t = -2.121320344 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = -2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -2.121320344 + -1.5 t = -2.121320344 + -1.5 Combine like terms: -2.121320344 + -1.5 = -3.621320344 t = -3.621320344 Simplifying t = -3.621320344Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.621320344, -3.621320344}Solution
t = {0.621320344, -3.621320344}
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