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Simplifying 0 = -16t2 + 48t + 12 Reorder the terms: 0 = 12 + 48t + -16t2 Solving 0 = 12 + 48t + -16t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -48t + 16t2 = 12 + 48t + -16t2 + -12 + -48t + 16t2 Reorder the terms: -12 + -48t + 16t2 = 12 + -12 + 48t + -48t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -12 + -48t + 16t2 = 0 + 48t + -48t + -16t2 + 16t2 -12 + -48t + 16t2 = 48t + -48t + -16t2 + 16t2 Combine like terms: 48t + -48t = 0 -12 + -48t + 16t2 = 0 + -16t2 + 16t2 -12 + -48t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12 + -48t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -12t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -12t + 4t2)' equal to zero and attempt to solve: Simplifying -3 + -12t + 4t2 = 0 Solving -3 + -12t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -3t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -3t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -3t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -3t + t2 = 0 + 0.75 -3t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -3t + t2 = 0.75 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 = 0.75 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 0.75 + 2.25 Combine like terms: 0.75 + 2.25 = 3 2.25 + -3t + t2 = 3 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (t + -1.5) equal to 1.732050808 and -1.732050808.Subproblem 1
t + -1.5 = 1.732050808 Simplifying t + -1.5 = 1.732050808 Reorder the terms: -1.5 + t = 1.732050808 Solving -1.5 + t = 1.732050808 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 = 1.732050808 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.732050808 + 1.5 t = 1.732050808 + 1.5 Combine like terms: 1.732050808 + 1.5 = 3.232050808 t = 3.232050808 Simplifying t = 3.232050808Subproblem 2
t + -1.5 = -1.732050808 Simplifying t + -1.5 = -1.732050808 Reorder the terms: -1.5 + t = -1.732050808 Solving -1.5 + t = -1.732050808 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 = -1.732050808 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.732050808 + 1.5 t = -1.732050808 + 1.5 Combine like terms: -1.732050808 + 1.5 = -0.232050808 t = -0.232050808 Simplifying t = -0.232050808Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.232050808, -0.232050808}Solution
t = {3.232050808, -0.232050808}
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