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Simplifying 2t2 + 1 = 6t Reorder the terms: 1 + 2t2 = 6t Solving 1 + 2t2 = 6t Solving for variable 't'. Reorder the terms: 1 + -6t + 2t2 = 6t + -6t Combine like terms: 6t + -6t = 0 1 + -6t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -3t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -3t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -3t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -3t + t2 = 0 + -0.5 -3t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -3t + t2 = -0.5 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = -0.5 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -0.5 + 2.25 Combine like terms: -0.5 + 2.25 = 1.75 2.25 + -3t + t2 = 1.75 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (t + -1.5) equal to 1.322875656 and -1.322875656.Subproblem 1
t + -1.5 = 1.322875656 Simplifying t + -1.5 = 1.322875656 Reorder the terms: -1.5 + t = 1.322875656 Solving -1.5 + t = 1.322875656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 1.322875656 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.322875656 + 1.5 t = 1.322875656 + 1.5 Combine like terms: 1.322875656 + 1.5 = 2.822875656 t = 2.822875656 Simplifying t = 2.822875656Subproblem 2
t + -1.5 = -1.322875656 Simplifying t + -1.5 = -1.322875656 Reorder the terms: -1.5 + t = -1.322875656 Solving -1.5 + t = -1.322875656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -1.322875656 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.322875656 + 1.5 t = -1.322875656 + 1.5 Combine like terms: -1.322875656 + 1.5 = 0.177124344 t = 0.177124344 Simplifying t = 0.177124344Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.822875656, 0.177124344}
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