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Simplifying 0 = 16t2 + -12t + -3 Reorder the terms: 0 = -3 + -12t + 16t2 Solving 0 = -3 + -12t + 16t2 Solving for variable 't'. Combine like terms: 0 + 3 = 3 3 + 12t + -16t2 = -3 + -12t + 16t2 + 3 + 12t + -16t2 Reorder the terms: 3 + 12t + -16t2 = -3 + 3 + -12t + 12t + 16t2 + -16t2 Combine like terms: -3 + 3 = 0 3 + 12t + -16t2 = 0 + -12t + 12t + 16t2 + -16t2 3 + 12t + -16t2 = -12t + 12t + 16t2 + -16t2 Combine like terms: -12t + 12t = 0 3 + 12t + -16t2 = 0 + 16t2 + -16t2 3 + 12t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 3 + 12t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.1875 + -0.75t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -0.75t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -0.75t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -0.75t + t2 = 0 + 0.1875 -0.75t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -0.75t + t2 = 0.1875 The t term is -0.75t. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75t + 0.140625 + t2 = 0.1875 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 0.1875 + 0.140625 Combine like terms: 0.1875 + 0.140625 = 0.328125 0.140625 + -0.75t + t2 = 0.328125 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 0.328125 Calculate the square root of the right side: 0.572821962 Break this problem into two subproblems by setting (t + -0.375) equal to 0.572821962 and -0.572821962.Subproblem 1
t + -0.375 = 0.572821962 Simplifying t + -0.375 = 0.572821962 Reorder the terms: -0.375 + t = 0.572821962 Solving -0.375 + t = 0.572821962 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = 0.572821962 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 0.572821962 + 0.375 t = 0.572821962 + 0.375 Combine like terms: 0.572821962 + 0.375 = 0.947821962 t = 0.947821962 Simplifying t = 0.947821962Subproblem 2
t + -0.375 = -0.572821962 Simplifying t + -0.375 = -0.572821962 Reorder the terms: -0.375 + t = -0.572821962 Solving -0.375 + t = -0.572821962 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = -0.572821962 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -0.572821962 + 0.375 t = -0.572821962 + 0.375 Combine like terms: -0.572821962 + 0.375 = -0.197821962 t = -0.197821962 Simplifying t = -0.197821962Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.947821962, -0.197821962}
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