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Simplifying 5 = -16t2 + 25t + 10 Reorder the terms: 5 = 10 + 25t + -16t2 Solving 5 = 10 + 25t + -16t2 Solving for variable 't'. Combine like terms: 5 + -10 = -5 -5 + -25t + 16t2 = 10 + 25t + -16t2 + -10 + -25t + 16t2 Reorder the terms: -5 + -25t + 16t2 = 10 + -10 + 25t + -25t + -16t2 + 16t2 Combine like terms: 10 + -10 = 0 -5 + -25t + 16t2 = 0 + 25t + -25t + -16t2 + 16t2 -5 + -25t + 16t2 = 25t + -25t + -16t2 + 16t2 Combine like terms: 25t + -25t = 0 -5 + -25t + 16t2 = 0 + -16t2 + 16t2 -5 + -25t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -25t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -1.5625t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -1.5625t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -1.5625t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -1.5625t + t2 = 0 + 0.3125 -1.5625t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -1.5625t + t2 = 0.3125 The t term is -1.5625t. Take half its coefficient (-0.78125). Square it (0.6103515625) and add it to both sides. Add '0.6103515625' to each side of the equation. -1.5625t + 0.6103515625 + t2 = 0.3125 + 0.6103515625 Reorder the terms: 0.6103515625 + -1.5625t + t2 = 0.3125 + 0.6103515625 Combine like terms: 0.3125 + 0.6103515625 = 0.9228515625 0.6103515625 + -1.5625t + t2 = 0.9228515625 Factor a perfect square on the left side: (t + -0.78125)(t + -0.78125) = 0.9228515625 Calculate the square root of the right side: 0.960651634 Break this problem into two subproblems by setting (t + -0.78125) equal to 0.960651634 and -0.960651634.Subproblem 1
t + -0.78125 = 0.960651634 Simplifying t + -0.78125 = 0.960651634 Reorder the terms: -0.78125 + t = 0.960651634 Solving -0.78125 + t = 0.960651634 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.78125' to each side of the equation. -0.78125 + 0.78125 + t = 0.960651634 + 0.78125 Combine like terms: -0.78125 + 0.78125 = 0.00000 0.00000 + t = 0.960651634 + 0.78125 t = 0.960651634 + 0.78125 Combine like terms: 0.960651634 + 0.78125 = 1.741901634 t = 1.741901634 Simplifying t = 1.741901634Subproblem 2
t + -0.78125 = -0.960651634 Simplifying t + -0.78125 = -0.960651634 Reorder the terms: -0.78125 + t = -0.960651634 Solving -0.78125 + t = -0.960651634 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.78125' to each side of the equation. -0.78125 + 0.78125 + t = -0.960651634 + 0.78125 Combine like terms: -0.78125 + 0.78125 = 0.00000 0.00000 + t = -0.960651634 + 0.78125 t = -0.960651634 + 0.78125 Combine like terms: -0.960651634 + 0.78125 = -0.179401634 t = -0.179401634 Simplifying t = -0.179401634Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.741901634, -0.179401634}
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