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Simplifying t = -16t2 + 25t + 5 Reorder the terms: t = 5 + 25t + -16t2 Solving t = 5 + 25t + -16t2 Solving for variable 't'. Reorder the terms: -5 + t + -25t + 16t2 = 5 + 25t + -16t2 + -5 + -25t + 16t2 Combine like terms: t + -25t = -24t -5 + -24t + 16t2 = 5 + 25t + -16t2 + -5 + -25t + 16t2 Reorder the terms: -5 + -24t + 16t2 = 5 + -5 + 25t + -25t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -24t + 16t2 = 0 + 25t + -25t + -16t2 + 16t2 -5 + -24t + 16t2 = 25t + -25t + -16t2 + 16t2 Combine like terms: 25t + -25t = 0 -5 + -24t + 16t2 = 0 + -16t2 + 16t2 -5 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -24t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -1.5t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -1.5t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -1.5t + t2 = 0 + 0.3125 -1.5t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -1.5t + t2 = 0.3125 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 0.3125 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.3125 + 0.5625 Combine like terms: 0.3125 + 0.5625 = 0.875 0.5625 + -1.5t + t2 = 0.875 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.875 Calculate the square root of the right side: 0.935414347 Break this problem into two subproblems by setting (t + -0.75) equal to 0.935414347 and -0.935414347.Subproblem 1
t + -0.75 = 0.935414347 Simplifying t + -0.75 = 0.935414347 Reorder the terms: -0.75 + t = 0.935414347 Solving -0.75 + t = 0.935414347 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 0.935414347 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.935414347 + 0.75 t = 0.935414347 + 0.75 Combine like terms: 0.935414347 + 0.75 = 1.685414347 t = 1.685414347 Simplifying t = 1.685414347Subproblem 2
t + -0.75 = -0.935414347 Simplifying t + -0.75 = -0.935414347 Reorder the terms: -0.75 + t = -0.935414347 Solving -0.75 + t = -0.935414347 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -0.935414347 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.935414347 + 0.75 t = -0.935414347 + 0.75 Combine like terms: -0.935414347 + 0.75 = -0.185414347 t = -0.185414347 Simplifying t = -0.185414347Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.685414347, -0.185414347}
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