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Simplifying 4(t) = -16t2 + 80t + 4 Reorder the terms: 4t = 4 + 80t + -16t2 Solving 4t = 4 + 80t + -16t2 Solving for variable 't'. Reorder the terms: -4 + 4t + -80t + 16t2 = 4 + 80t + -16t2 + -4 + -80t + 16t2 Combine like terms: 4t + -80t = -76t -4 + -76t + 16t2 = 4 + 80t + -16t2 + -4 + -80t + 16t2 Reorder the terms: -4 + -76t + 16t2 = 4 + -4 + 80t + -80t + -16t2 + 16t2 Combine like terms: 4 + -4 = 0 -4 + -76t + 16t2 = 0 + 80t + -80t + -16t2 + 16t2 -4 + -76t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 -4 + -76t + 16t2 = 0 + -16t2 + 16t2 -4 + -76t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -4 + -76t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-1 + -19t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1 + -19t + 4t2)' equal to zero and attempt to solve: Simplifying -1 + -19t + 4t2 = 0 Solving -1 + -19t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -4.75t + t2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -4.75t + 0.25 + t2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -4.75t + t2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -4.75t + t2 = 0 + 0.25 -4.75t + t2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -4.75t + t2 = 0.25 The t term is -4.75t. Take half its coefficient (-2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. -4.75t + 5.640625 + t2 = 0.25 + 5.640625 Reorder the terms: 5.640625 + -4.75t + t2 = 0.25 + 5.640625 Combine like terms: 0.25 + 5.640625 = 5.890625 5.640625 + -4.75t + t2 = 5.890625 Factor a perfect square on the left side: (t + -2.375)(t + -2.375) = 5.890625 Calculate the square root of the right side: 2.42706098 Break this problem into two subproblems by setting (t + -2.375) equal to 2.42706098 and -2.42706098.Subproblem 1
t + -2.375 = 2.42706098 Simplifying t + -2.375 = 2.42706098 Reorder the terms: -2.375 + t = 2.42706098 Solving -2.375 + t = 2.42706098 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + t = 2.42706098 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + t = 2.42706098 + 2.375 t = 2.42706098 + 2.375 Combine like terms: 2.42706098 + 2.375 = 4.80206098 t = 4.80206098 Simplifying t = 4.80206098Subproblem 2
t + -2.375 = -2.42706098 Simplifying t + -2.375 = -2.42706098 Reorder the terms: -2.375 + t = -2.42706098 Solving -2.375 + t = -2.42706098 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + t = -2.42706098 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + t = -2.42706098 + 2.375 t = -2.42706098 + 2.375 Combine like terms: -2.42706098 + 2.375 = -0.05206098 t = -0.05206098 Simplifying t = -0.05206098Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.80206098, -0.05206098}Solution
t = {4.80206098, -0.05206098}
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