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Simplifying 10 = -16t2 + 20t + 5 Reorder the terms: 10 = 5 + 20t + -16t2 Solving 10 = 5 + 20t + -16t2 Solving for variable 't'. Combine like terms: 10 + -5 = 5 5 + -20t + 16t2 = 5 + 20t + -16t2 + -5 + -20t + 16t2 Reorder the terms: 5 + -20t + 16t2 = 5 + -5 + 20t + -20t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 5 + -20t + 16t2 = 0 + 20t + -20t + -16t2 + 16t2 5 + -20t + 16t2 = 20t + -20t + -16t2 + 16t2 Combine like terms: 20t + -20t = 0 5 + -20t + 16t2 = 0 + -16t2 + 16t2 5 + -20t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 5 + -20t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + -1.25t + t2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + -1.25t + -0.3125 + t2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + -1.25t + t2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + -1.25t + t2 = 0 + -0.3125 -1.25t + t2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 -1.25t + t2 = -0.3125 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 = -0.3125 + 0.390625 Reorder the terms: 0.390625 + -1.25t + t2 = -0.3125 + 0.390625 Combine like terms: -0.3125 + 0.390625 = 0.078125 0.390625 + -1.25t + t2 = 0.078125 Factor a perfect square on the left side: (t + -0.625)(t + -0.625) = 0.078125 Calculate the square root of the right side: 0.279508497 Break this problem into two subproblems by setting (t + -0.625) equal to 0.279508497 and -0.279508497.Subproblem 1
t + -0.625 = 0.279508497 Simplifying t + -0.625 = 0.279508497 Reorder the terms: -0.625 + t = 0.279508497 Solving -0.625 + t = 0.279508497 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 = 0.279508497 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = 0.279508497 + 0.625 t = 0.279508497 + 0.625 Combine like terms: 0.279508497 + 0.625 = 0.904508497 t = 0.904508497 Simplifying t = 0.904508497Subproblem 2
t + -0.625 = -0.279508497 Simplifying t + -0.625 = -0.279508497 Reorder the terms: -0.625 + t = -0.279508497 Solving -0.625 + t = -0.279508497 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 = -0.279508497 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = -0.279508497 + 0.625 t = -0.279508497 + 0.625 Combine like terms: -0.279508497 + 0.625 = 0.345491503 t = 0.345491503 Simplifying t = 0.345491503Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.904508497, 0.345491503}
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