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Simplifying 0 = -16t2 + 128t + 12 Reorder the terms: 0 = 12 + 128t + -16t2 Solving 0 = 12 + 128t + -16t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -128t + 16t2 = 12 + 128t + -16t2 + -12 + -128t + 16t2 Reorder the terms: -12 + -128t + 16t2 = 12 + -12 + 128t + -128t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -12 + -128t + 16t2 = 0 + 128t + -128t + -16t2 + 16t2 -12 + -128t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 -12 + -128t + 16t2 = 0 + -16t2 + 16t2 -12 + -128t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12 + -128t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -32t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -32t + 4t2)' equal to zero and attempt to solve: Simplifying -3 + -32t + 4t2 = 0 Solving -3 + -32t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -8t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -8t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -8t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -8t + t2 = 0 + 0.75 -8t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -8t + t2 = 0.75 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = 0.75 + 16 Reorder the terms: 16 + -8t + t2 = 0.75 + 16 Combine like terms: 0.75 + 16 = 16.75 16 + -8t + t2 = 16.75 Factor a perfect square on the left side: (t + -4)(t + -4) = 16.75 Calculate the square root of the right side: 4.092676386 Break this problem into two subproblems by setting (t + -4) equal to 4.092676386 and -4.092676386.Subproblem 1
t + -4 = 4.092676386 Simplifying t + -4 = 4.092676386 Reorder the terms: -4 + t = 4.092676386 Solving -4 + t = 4.092676386 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 4.092676386 + 4 Combine like terms: -4 + 4 = 0 0 + t = 4.092676386 + 4 t = 4.092676386 + 4 Combine like terms: 4.092676386 + 4 = 8.092676386 t = 8.092676386 Simplifying t = 8.092676386Subproblem 2
t + -4 = -4.092676386 Simplifying t + -4 = -4.092676386 Reorder the terms: -4 + t = -4.092676386 Solving -4 + t = -4.092676386 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -4.092676386 + 4 Combine like terms: -4 + 4 = 0 0 + t = -4.092676386 + 4 t = -4.092676386 + 4 Combine like terms: -4.092676386 + 4 = -0.092676386 t = -0.092676386 Simplifying t = -0.092676386Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.092676386, -0.092676386}Solution
t = {8.092676386, -0.092676386}
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