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Simplifying 2.5 = -16t2 + 45t + 5.5 Reorder the terms: 2.5 = 5.5 + 45t + -16t2 Solving 2.5 = 5.5 + 45t + -16t2 Solving for variable 't'. Combine like terms: 2.5 + -5.5 = -3 -3 + -45t + 16t2 = 5.5 + 45t + -16t2 + -5.5 + -45t + 16t2 Reorder the terms: -3 + -45t + 16t2 = 5.5 + -5.5 + 45t + -45t + -16t2 + 16t2 Combine like terms: 5.5 + -5.5 = 0.0 -3 + -45t + 16t2 = 0.0 + 45t + -45t + -16t2 + 16t2 -3 + -45t + 16t2 = 45t + -45t + -16t2 + 16t2 Combine like terms: 45t + -45t = 0 -3 + -45t + 16t2 = 0 + -16t2 + 16t2 -3 + -45t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3 + -45t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -2.8125t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -2.8125t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -2.8125t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -2.8125t + t2 = 0 + 0.1875 -2.8125t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -2.8125t + t2 = 0.1875 The t term is -2.8125t. Take half its coefficient (-1.40625). Square it (1.977539063) and add it to both sides. Add '1.977539063' to each side of the equation. -2.8125t + 1.977539063 + t2 = 0.1875 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125t + t2 = 0.1875 + 1.977539063 Combine like terms: 0.1875 + 1.977539063 = 2.165039063 1.977539063 + -2.8125t + t2 = 2.165039063 Factor a perfect square on the left side: (t + -1.40625)(t + -1.40625) = 2.165039063 Calculate the square root of the right side: 1.471407171 Break this problem into two subproblems by setting (t + -1.40625) equal to 1.471407171 and -1.471407171.Subproblem 1
t + -1.40625 = 1.471407171 Simplifying t + -1.40625 = 1.471407171 Reorder the terms: -1.40625 + t = 1.471407171 Solving -1.40625 + t = 1.471407171 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = 1.471407171 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = 1.471407171 + 1.40625 t = 1.471407171 + 1.40625 Combine like terms: 1.471407171 + 1.40625 = 2.877657171 t = 2.877657171 Simplifying t = 2.877657171Subproblem 2
t + -1.40625 = -1.471407171 Simplifying t + -1.40625 = -1.471407171 Reorder the terms: -1.40625 + t = -1.471407171 Solving -1.40625 + t = -1.471407171 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = -1.471407171 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = -1.471407171 + 1.40625 t = -1.471407171 + 1.40625 Combine like terms: -1.471407171 + 1.40625 = -0.065157171 t = -0.065157171 Simplifying t = -0.065157171Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.877657171, -0.065157171}
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