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Simplifying 180 = 16t2 + 80t + 80 Reorder the terms: 180 = 80 + 80t + 16t2 Solving 180 = 80 + 80t + 16t2 Solving for variable 't'. Combine like terms: 180 + -80 = 100 100 + -80t + -16t2 = 80 + 80t + 16t2 + -80 + -80t + -16t2 Reorder the terms: 100 + -80t + -16t2 = 80 + -80 + 80t + -80t + 16t2 + -16t2 Combine like terms: 80 + -80 = 0 100 + -80t + -16t2 = 0 + 80t + -80t + 16t2 + -16t2 100 + -80t + -16t2 = 80t + -80t + 16t2 + -16t2 Combine like terms: 80t + -80t = 0 100 + -80t + -16t2 = 0 + 16t2 + -16t2 100 + -80t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 100 + -80t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(25 + -20t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(25 + -20t + -4t2)' equal to zero and attempt to solve: Simplifying 25 + -20t + -4t2 = 0 Solving 25 + -20t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -6.25 + 5t + t2 = 0 Move the constant term to the right: Add '6.25' to each side of the equation. -6.25 + 5t + 6.25 + t2 = 0 + 6.25 Reorder the terms: -6.25 + 6.25 + 5t + t2 = 0 + 6.25 Combine like terms: -6.25 + 6.25 = 0.00 0.00 + 5t + t2 = 0 + 6.25 5t + t2 = 0 + 6.25 Combine like terms: 0 + 6.25 = 6.25 5t + t2 = 6.25 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = 6.25 + 6.25 Reorder the terms: 6.25 + 5t + t2 = 6.25 + 6.25 Combine like terms: 6.25 + 6.25 = 12.5 6.25 + 5t + t2 = 12.5 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 12.5 Calculate the square root of the right side: 3.535533906 Break this problem into two subproblems by setting (t + 2.5) equal to 3.535533906 and -3.535533906.Subproblem 1
t + 2.5 = 3.535533906 Simplifying t + 2.5 = 3.535533906 Reorder the terms: 2.5 + t = 3.535533906 Solving 2.5 + t = 3.535533906 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 3.535533906 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 3.535533906 + -2.5 t = 3.535533906 + -2.5 Combine like terms: 3.535533906 + -2.5 = 1.035533906 t = 1.035533906 Simplifying t = 1.035533906Subproblem 2
t + 2.5 = -3.535533906 Simplifying t + 2.5 = -3.535533906 Reorder the terms: 2.5 + t = -3.535533906 Solving 2.5 + t = -3.535533906 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -3.535533906 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -3.535533906 + -2.5 t = -3.535533906 + -2.5 Combine like terms: -3.535533906 + -2.5 = -6.035533906 t = -6.035533906 Simplifying t = -6.035533906Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.035533906, -6.035533906}Solution
t = {1.035533906, -6.035533906}
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