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Simplifying 0 = -16t2 + 81t + 5 Reorder the terms: 0 = 5 + 81t + -16t2 Solving 0 = 5 + 81t + -16t2 Solving for variable 't'. Combine like terms: 0 + -5 = -5 -5 + -81t + 16t2 = 5 + 81t + -16t2 + -5 + -81t + 16t2 Reorder the terms: -5 + -81t + 16t2 = 5 + -5 + 81t + -81t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -81t + 16t2 = 0 + 81t + -81t + -16t2 + 16t2 -5 + -81t + 16t2 = 81t + -81t + -16t2 + 16t2 Combine like terms: 81t + -81t = 0 -5 + -81t + 16t2 = 0 + -16t2 + 16t2 -5 + -81t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -81t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -5.0625t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -5.0625t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -5.0625t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -5.0625t + t2 = 0 + 0.3125 -5.0625t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -5.0625t + t2 = 0.3125 The t term is -5.0625t. Take half its coefficient (-2.53125). Square it (6.407226563) and add it to both sides. Add '6.407226563' to each side of the equation. -5.0625t + 6.407226563 + t2 = 0.3125 + 6.407226563 Reorder the terms: 6.407226563 + -5.0625t + t2 = 0.3125 + 6.407226563 Combine like terms: 0.3125 + 6.407226563 = 6.719726563 6.407226563 + -5.0625t + t2 = 6.719726563 Factor a perfect square on the left side: (t + -2.53125)(t + -2.53125) = 6.719726563 Calculate the square root of the right side: 2.592243539 Break this problem into two subproblems by setting (t + -2.53125) equal to 2.592243539 and -2.592243539.Subproblem 1
t + -2.53125 = 2.592243539 Simplifying t + -2.53125 = 2.592243539 Reorder the terms: -2.53125 + t = 2.592243539 Solving -2.53125 + t = 2.592243539 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.53125' to each side of the equation. -2.53125 + 2.53125 + t = 2.592243539 + 2.53125 Combine like terms: -2.53125 + 2.53125 = 0.00000 0.00000 + t = 2.592243539 + 2.53125 t = 2.592243539 + 2.53125 Combine like terms: 2.592243539 + 2.53125 = 5.123493539 t = 5.123493539 Simplifying t = 5.123493539Subproblem 2
t + -2.53125 = -2.592243539 Simplifying t + -2.53125 = -2.592243539 Reorder the terms: -2.53125 + t = -2.592243539 Solving -2.53125 + t = -2.592243539 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.53125' to each side of the equation. -2.53125 + 2.53125 + t = -2.592243539 + 2.53125 Combine like terms: -2.53125 + 2.53125 = 0.00000 0.00000 + t = -2.592243539 + 2.53125 t = -2.592243539 + 2.53125 Combine like terms: -2.592243539 + 2.53125 = -0.060993539 t = -0.060993539 Simplifying t = -0.060993539Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.123493539, -0.060993539}
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