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Simplifying 22 = 3 + 40t + -16t2 Solving 22 = 3 + 40t + -16t2 Solving for variable 't'. Combine like terms: 22 + -3 = 19 19 + -40t + 16t2 = 3 + 40t + -16t2 + -3 + -40t + 16t2 Reorder the terms: 19 + -40t + 16t2 = 3 + -3 + 40t + -40t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 19 + -40t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 19 + -40t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 19 + -40t + 16t2 = 0 + -16t2 + 16t2 19 + -40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 19 + -40t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.1875 + -2.5t + t2 = 0 Move the constant term to the right: Add '-1.1875' to each side of the equation. 1.1875 + -2.5t + -1.1875 + t2 = 0 + -1.1875 Reorder the terms: 1.1875 + -1.1875 + -2.5t + t2 = 0 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + -2.5t + t2 = 0 + -1.1875 -2.5t + t2 = 0 + -1.1875 Combine like terms: 0 + -1.1875 = -1.1875 -2.5t + t2 = -1.1875 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = -1.1875 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = -1.1875 + 1.5625 Combine like terms: -1.1875 + 1.5625 = 0.375 1.5625 + -2.5t + t2 = 0.375 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (t + -1.25) equal to 0.612372436 and -0.612372436.Subproblem 1
t + -1.25 = 0.612372436 Simplifying t + -1.25 = 0.612372436 Reorder the terms: -1.25 + t = 0.612372436 Solving -1.25 + t = 0.612372436 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 0.612372436 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 0.612372436 + 1.25 t = 0.612372436 + 1.25 Combine like terms: 0.612372436 + 1.25 = 1.862372436 t = 1.862372436 Simplifying t = 1.862372436Subproblem 2
t + -1.25 = -0.612372436 Simplifying t + -1.25 = -0.612372436 Reorder the terms: -1.25 + t = -0.612372436 Solving -1.25 + t = -0.612372436 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -0.612372436 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -0.612372436 + 1.25 t = -0.612372436 + 1.25 Combine like terms: -0.612372436 + 1.25 = 0.637627564 t = 0.637627564 Simplifying t = 0.637627564Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.862372436, 0.637627564}
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