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Simplifying 110 = 80t + -5t2 Solving 110 = 80t + -5t2 Solving for variable 't'. Reorder the terms: 110 + -80t + 5t2 = 80t + -80t + -5t2 + 5t2 Combine like terms: 80t + -80t = 0 110 + -80t + 5t2 = 0 + -5t2 + 5t2 110 + -80t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 110 + -80t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(22 + -16t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(22 + -16t + t2)' equal to zero and attempt to solve: Simplifying 22 + -16t + t2 = 0 Solving 22 + -16t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-22' to each side of the equation. 22 + -16t + -22 + t2 = 0 + -22 Reorder the terms: 22 + -22 + -16t + t2 = 0 + -22 Combine like terms: 22 + -22 = 0 0 + -16t + t2 = 0 + -22 -16t + t2 = 0 + -22 Combine like terms: 0 + -22 = -22 -16t + t2 = -22 The t term is -16t. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16t + 64 + t2 = -22 + 64 Reorder the terms: 64 + -16t + t2 = -22 + 64 Combine like terms: -22 + 64 = 42 64 + -16t + t2 = 42 Factor a perfect square on the left side: (t + -8)(t + -8) = 42 Calculate the square root of the right side: 6.480740698 Break this problem into two subproblems by setting (t + -8) equal to 6.480740698 and -6.480740698.Subproblem 1
t + -8 = 6.480740698 Simplifying t + -8 = 6.480740698 Reorder the terms: -8 + t = 6.480740698 Solving -8 + t = 6.480740698 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = 6.480740698 + 8 Combine like terms: -8 + 8 = 0 0 + t = 6.480740698 + 8 t = 6.480740698 + 8 Combine like terms: 6.480740698 + 8 = 14.480740698 t = 14.480740698 Simplifying t = 14.480740698Subproblem 2
t + -8 = -6.480740698 Simplifying t + -8 = -6.480740698 Reorder the terms: -8 + t = -6.480740698 Solving -8 + t = -6.480740698 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = -6.480740698 + 8 Combine like terms: -8 + 8 = 0 0 + t = -6.480740698 + 8 t = -6.480740698 + 8 Combine like terms: -6.480740698 + 8 = 1.519259302 t = 1.519259302 Simplifying t = 1.519259302Solution
The solution to the problem is based on the solutions from the subproblems. t = {14.480740698, 1.519259302}Solution
t = {14.480740698, 1.519259302}
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