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Simplifying 45t = -16t2 + 25t + 45 Reorder the terms: 45t = 45 + 25t + -16t2 Solving 45t = 45 + 25t + -16t2 Solving for variable 't'. Reorder the terms: -45 + 45t + -25t + 16t2 = 45 + 25t + -16t2 + -45 + -25t + 16t2 Combine like terms: 45t + -25t = 20t -45 + 20t + 16t2 = 45 + 25t + -16t2 + -45 + -25t + 16t2 Reorder the terms: -45 + 20t + 16t2 = 45 + -45 + 25t + -25t + -16t2 + 16t2 Combine like terms: 45 + -45 = 0 -45 + 20t + 16t2 = 0 + 25t + -25t + -16t2 + 16t2 -45 + 20t + 16t2 = 25t + -25t + -16t2 + 16t2 Combine like terms: 25t + -25t = 0 -45 + 20t + 16t2 = 0 + -16t2 + 16t2 -45 + 20t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -45 + 20t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2.8125 + 1.25t + t2 = 0 Move the constant term to the right: Add '2.8125' to each side of the equation. -2.8125 + 1.25t + 2.8125 + t2 = 0 + 2.8125 Reorder the terms: -2.8125 + 2.8125 + 1.25t + t2 = 0 + 2.8125 Combine like terms: -2.8125 + 2.8125 = 0.0000 0.0000 + 1.25t + t2 = 0 + 2.8125 1.25t + t2 = 0 + 2.8125 Combine like terms: 0 + 2.8125 = 2.8125 1.25t + t2 = 2.8125 The t term is 1.25t. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25t + 0.390625 + t2 = 2.8125 + 0.390625 Reorder the terms: 0.390625 + 1.25t + t2 = 2.8125 + 0.390625 Combine like terms: 2.8125 + 0.390625 = 3.203125 0.390625 + 1.25t + t2 = 3.203125 Factor a perfect square on the left side: (t + 0.625)(t + 0.625) = 3.203125 Calculate the square root of the right side: 1.789727633 Break this problem into two subproblems by setting (t + 0.625) equal to 1.789727633 and -1.789727633.Subproblem 1
t + 0.625 = 1.789727633 Simplifying t + 0.625 = 1.789727633 Reorder the terms: 0.625 + t = 1.789727633 Solving 0.625 + t = 1.789727633 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = 1.789727633 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = 1.789727633 + -0.625 t = 1.789727633 + -0.625 Combine like terms: 1.789727633 + -0.625 = 1.164727633 t = 1.164727633 Simplifying t = 1.164727633Subproblem 2
t + 0.625 = -1.789727633 Simplifying t + 0.625 = -1.789727633 Reorder the terms: 0.625 + t = -1.789727633 Solving 0.625 + t = -1.789727633 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = -1.789727633 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = -1.789727633 + -0.625 t = -1.789727633 + -0.625 Combine like terms: -1.789727633 + -0.625 = -2.414727633 t = -2.414727633 Simplifying t = -2.414727633Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.164727633, -2.414727633}
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