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Simplifying 450 = 16t2 + 80t + 401 Reorder the terms: 450 = 401 + 80t + 16t2 Solving 450 = 401 + 80t + 16t2 Solving for variable 't'. Combine like terms: 450 + -401 = 49 49 + -80t + -16t2 = 401 + 80t + 16t2 + -401 + -80t + -16t2 Reorder the terms: 49 + -80t + -16t2 = 401 + -401 + 80t + -80t + 16t2 + -16t2 Combine like terms: 401 + -401 = 0 49 + -80t + -16t2 = 0 + 80t + -80t + 16t2 + -16t2 49 + -80t + -16t2 = 80t + -80t + 16t2 + -16t2 Combine like terms: 80t + -80t = 0 49 + -80t + -16t2 = 0 + 16t2 + -16t2 49 + -80t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 49 + -80t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -3.0625 + 5t + t2 = 0 Move the constant term to the right: Add '3.0625' to each side of the equation. -3.0625 + 5t + 3.0625 + t2 = 0 + 3.0625 Reorder the terms: -3.0625 + 3.0625 + 5t + t2 = 0 + 3.0625 Combine like terms: -3.0625 + 3.0625 = 0.0000 0.0000 + 5t + t2 = 0 + 3.0625 5t + t2 = 0 + 3.0625 Combine like terms: 0 + 3.0625 = 3.0625 5t + t2 = 3.0625 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 = 3.0625 + 6.25 Reorder the terms: 6.25 + 5t + t2 = 3.0625 + 6.25 Combine like terms: 3.0625 + 6.25 = 9.3125 6.25 + 5t + t2 = 9.3125 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 9.3125 Calculate the square root of the right side: 3.051638904 Break this problem into two subproblems by setting (t + 2.5) equal to 3.051638904 and -3.051638904.Subproblem 1
t + 2.5 = 3.051638904 Simplifying t + 2.5 = 3.051638904 Reorder the terms: 2.5 + t = 3.051638904 Solving 2.5 + t = 3.051638904 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 = 3.051638904 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 3.051638904 + -2.5 t = 3.051638904 + -2.5 Combine like terms: 3.051638904 + -2.5 = 0.551638904 t = 0.551638904 Simplifying t = 0.551638904Subproblem 2
t + 2.5 = -3.051638904 Simplifying t + 2.5 = -3.051638904 Reorder the terms: 2.5 + t = -3.051638904 Solving 2.5 + t = -3.051638904 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 = -3.051638904 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -3.051638904 + -2.5 t = -3.051638904 + -2.5 Combine like terms: -3.051638904 + -2.5 = -5.551638904 t = -5.551638904 Simplifying t = -5.551638904Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.551638904, -5.551638904}
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