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Simplifying 0.75 = 4t(-4t + 6) Reorder the terms: 0.75 = 4t(6 + -4t) 0.75 = (6 * 4t + -4t * 4t) 0.75 = (24t + -16t2) Solving 0.75 = 24t + -16t2 Solving for variable 't'. Reorder the terms: 0.75 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 0.75 + -24t + 16t2 = 0 + -16t2 + 16t2 0.75 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 0.75 + -24t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.046875 + -1.5t + t2 = 0 Move the constant term to the right: Add '-0.046875' to each side of the equation. 0.046875 + -1.5t + -0.046875 + t2 = 0 + -0.046875 Reorder the terms: 0.046875 + -0.046875 + -1.5t + t2 = 0 + -0.046875 Combine like terms: 0.046875 + -0.046875 = 0.000000 0.000000 + -1.5t + t2 = 0 + -0.046875 -1.5t + t2 = 0 + -0.046875 Combine like terms: 0 + -0.046875 = -0.046875 -1.5t + t2 = -0.046875 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 = -0.046875 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = -0.046875 + 0.5625 Combine like terms: -0.046875 + 0.5625 = 0.515625 0.5625 + -1.5t + t2 = 0.515625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (t + -0.75) equal to 0.718070331 and -0.718070331.Subproblem 1
t + -0.75 = 0.718070331 Simplifying t + -0.75 = 0.718070331 Reorder the terms: -0.75 + t = 0.718070331 Solving -0.75 + t = 0.718070331 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 = 0.718070331 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.718070331 + 0.75 t = 0.718070331 + 0.75 Combine like terms: 0.718070331 + 0.75 = 1.468070331 t = 1.468070331 Simplifying t = 1.468070331Subproblem 2
t + -0.75 = -0.718070331 Simplifying t + -0.75 = -0.718070331 Reorder the terms: -0.75 + t = -0.718070331 Solving -0.75 + t = -0.718070331 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 = -0.718070331 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.718070331 + 0.75 t = -0.718070331 + 0.75 Combine like terms: -0.718070331 + 0.75 = 0.031929669 t = 0.031929669 Simplifying t = 0.031929669Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.468070331, 0.031929669}
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