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Simplifying 75 = 2 + 90t + -16t2 Solving 75 = 2 + 90t + -16t2 Solving for variable 't'. Combine like terms: 75 + -2 = 73 73 + -90t + 16t2 = 2 + 90t + -16t2 + -2 + -90t + 16t2 Reorder the terms: 73 + -90t + 16t2 = 2 + -2 + 90t + -90t + -16t2 + 16t2 Combine like terms: 2 + -2 = 0 73 + -90t + 16t2 = 0 + 90t + -90t + -16t2 + 16t2 73 + -90t + 16t2 = 90t + -90t + -16t2 + 16t2 Combine like terms: 90t + -90t = 0 73 + -90t + 16t2 = 0 + -16t2 + 16t2 73 + -90t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 73 + -90t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 4.5625 + -5.625t + t2 = 0 Move the constant term to the right: Add '-4.5625' to each side of the equation. 4.5625 + -5.625t + -4.5625 + t2 = 0 + -4.5625 Reorder the terms: 4.5625 + -4.5625 + -5.625t + t2 = 0 + -4.5625 Combine like terms: 4.5625 + -4.5625 = 0.0000 0.0000 + -5.625t + t2 = 0 + -4.5625 -5.625t + t2 = 0 + -4.5625 Combine like terms: 0 + -4.5625 = -4.5625 -5.625t + t2 = -4.5625 The t term is -5.625t. Take half its coefficient (-2.8125). Square it (7.91015625) and add it to both sides. Add '7.91015625' to each side of the equation. -5.625t + 7.91015625 + t2 = -4.5625 + 7.91015625 Reorder the terms: 7.91015625 + -5.625t + t2 = -4.5625 + 7.91015625 Combine like terms: -4.5625 + 7.91015625 = 3.34765625 7.91015625 + -5.625t + t2 = 3.34765625 Factor a perfect square on the left side: (t + -2.8125)(t + -2.8125) = 3.34765625 Calculate the square root of the right side: 1.829660146 Break this problem into two subproblems by setting (t + -2.8125) equal to 1.829660146 and -1.829660146.Subproblem 1
t + -2.8125 = 1.829660146 Simplifying t + -2.8125 = 1.829660146 Reorder the terms: -2.8125 + t = 1.829660146 Solving -2.8125 + t = 1.829660146 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.8125' to each side of the equation. -2.8125 + 2.8125 + t = 1.829660146 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = 1.829660146 + 2.8125 t = 1.829660146 + 2.8125 Combine like terms: 1.829660146 + 2.8125 = 4.642160146 t = 4.642160146 Simplifying t = 4.642160146Subproblem 2
t + -2.8125 = -1.829660146 Simplifying t + -2.8125 = -1.829660146 Reorder the terms: -2.8125 + t = -1.829660146 Solving -2.8125 + t = -1.829660146 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.8125' to each side of the equation. -2.8125 + 2.8125 + t = -1.829660146 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + t = -1.829660146 + 2.8125 t = -1.829660146 + 2.8125 Combine like terms: -1.829660146 + 2.8125 = 0.982839854 t = 0.982839854 Simplifying t = 0.982839854Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.642160146, 0.982839854}
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