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Simplifying 0 = 16t2 + -24t + 7 Reorder the terms: 0 = 7 + -24t + 16t2 Solving 0 = 7 + -24t + 16t2 Solving for variable 't'. Combine like terms: 0 + -7 = -7 -7 + 24t + -16t2 = 7 + -24t + 16t2 + -7 + 24t + -16t2 Reorder the terms: -7 + 24t + -16t2 = 7 + -7 + -24t + 24t + 16t2 + -16t2 Combine like terms: 7 + -7 = 0 -7 + 24t + -16t2 = 0 + -24t + 24t + 16t2 + -16t2 -7 + 24t + -16t2 = -24t + 24t + 16t2 + -16t2 Combine like terms: -24t + 24t = 0 -7 + 24t + -16t2 = 0 + 16t2 + -16t2 -7 + 24t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -7 + 24t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.4375 + -1.5t + t2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + -1.5t + -0.4375 + t2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + -1.5t + t2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + -1.5t + t2 = 0 + -0.4375 -1.5t + t2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 -1.5t + t2 = -0.4375 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 = -0.4375 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = -0.4375 + 0.5625 Combine like terms: -0.4375 + 0.5625 = 0.125 0.5625 + -1.5t + t2 = 0.125 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (t + -0.75) equal to 0.353553391 and -0.353553391.Subproblem 1
t + -0.75 = 0.353553391 Simplifying t + -0.75 = 0.353553391 Reorder the terms: -0.75 + t = 0.353553391 Solving -0.75 + t = 0.353553391 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 = 0.353553391 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.353553391 + 0.75 t = 0.353553391 + 0.75 Combine like terms: 0.353553391 + 0.75 = 1.103553391 t = 1.103553391 Simplifying t = 1.103553391Subproblem 2
t + -0.75 = -0.353553391 Simplifying t + -0.75 = -0.353553391 Reorder the terms: -0.75 + t = -0.353553391 Solving -0.75 + t = -0.353553391 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 = -0.353553391 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.353553391 + 0.75 t = -0.353553391 + 0.75 Combine like terms: -0.353553391 + 0.75 = 0.396446609 t = 0.396446609 Simplifying t = 0.396446609Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.103553391, 0.396446609}
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