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Simplifying 16t2 + -40t + 3 = 0 Reorder the terms: 3 + -40t + 16t2 = 0 Solving 3 + -40t + 16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + -2.5t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -2.5t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -2.5t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -2.5t + t2 = 0 + -0.1875 -2.5t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -2.5t + t2 = -0.1875 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = -0.1875 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = -0.1875 + 1.5625 Combine like terms: -0.1875 + 1.5625 = 1.375 1.5625 + -2.5t + t2 = 1.375 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 1.375 Calculate the square root of the right side: 1.17260394 Break this problem into two subproblems by setting (t + -1.25) equal to 1.17260394 and -1.17260394.Subproblem 1
t + -1.25 = 1.17260394 Simplifying t + -1.25 = 1.17260394 Reorder the terms: -1.25 + t = 1.17260394 Solving -1.25 + t = 1.17260394 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 1.17260394 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 1.17260394 + 1.25 t = 1.17260394 + 1.25 Combine like terms: 1.17260394 + 1.25 = 2.42260394 t = 2.42260394 Simplifying t = 2.42260394Subproblem 2
t + -1.25 = -1.17260394 Simplifying t + -1.25 = -1.17260394 Reorder the terms: -1.25 + t = -1.17260394 Solving -1.25 + t = -1.17260394 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -1.17260394 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -1.17260394 + 1.25 t = -1.17260394 + 1.25 Combine like terms: -1.17260394 + 1.25 = 0.07739606 t = 0.07739606 Simplifying t = 0.07739606Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.42260394, 0.07739606}
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