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Simplifying 45(t) = -16t2 + 5t + 105 Reorder the terms: 45t = 105 + 5t + -16t2 Solving 45t = 105 + 5t + -16t2 Solving for variable 't'. Reorder the terms: -105 + 45t + -5t + 16t2 = 105 + 5t + -16t2 + -105 + -5t + 16t2 Combine like terms: 45t + -5t = 40t -105 + 40t + 16t2 = 105 + 5t + -16t2 + -105 + -5t + 16t2 Reorder the terms: -105 + 40t + 16t2 = 105 + -105 + 5t + -5t + -16t2 + 16t2 Combine like terms: 105 + -105 = 0 -105 + 40t + 16t2 = 0 + 5t + -5t + -16t2 + 16t2 -105 + 40t + 16t2 = 5t + -5t + -16t2 + 16t2 Combine like terms: 5t + -5t = 0 -105 + 40t + 16t2 = 0 + -16t2 + 16t2 -105 + 40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -105 + 40t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -6.5625 + 2.5t + t2 = 0 Move the constant term to the right: Add '6.5625' to each side of the equation. -6.5625 + 2.5t + 6.5625 + t2 = 0 + 6.5625 Reorder the terms: -6.5625 + 6.5625 + 2.5t + t2 = 0 + 6.5625 Combine like terms: -6.5625 + 6.5625 = 0.0000 0.0000 + 2.5t + t2 = 0 + 6.5625 2.5t + t2 = 0 + 6.5625 Combine like terms: 0 + 6.5625 = 6.5625 2.5t + t2 = 6.5625 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 = 6.5625 + 1.5625 Reorder the terms: 1.5625 + 2.5t + t2 = 6.5625 + 1.5625 Combine like terms: 6.5625 + 1.5625 = 8.125 1.5625 + 2.5t + t2 = 8.125 Factor a perfect square on the left side: (t + 1.25)(t + 1.25) = 8.125 Calculate the square root of the right side: 2.850438563 Break this problem into two subproblems by setting (t + 1.25) equal to 2.850438563 and -2.850438563.Subproblem 1
t + 1.25 = 2.850438563 Simplifying t + 1.25 = 2.850438563 Reorder the terms: 1.25 + t = 2.850438563 Solving 1.25 + t = 2.850438563 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 = 2.850438563 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = 2.850438563 + -1.25 t = 2.850438563 + -1.25 Combine like terms: 2.850438563 + -1.25 = 1.600438563 t = 1.600438563 Simplifying t = 1.600438563Subproblem 2
t + 1.25 = -2.850438563 Simplifying t + 1.25 = -2.850438563 Reorder the terms: 1.25 + t = -2.850438563 Solving 1.25 + t = -2.850438563 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 = -2.850438563 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = -2.850438563 + -1.25 t = -2.850438563 + -1.25 Combine like terms: -2.850438563 + -1.25 = -4.100438563 t = -4.100438563 Simplifying t = -4.100438563Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.600438563, -4.100438563}
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