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Simplifying 45t2 + -30t + 1 = 0 Reorder the terms: 1 + -30t + 45t2 = 0 Solving 1 + -30t + 45t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. 0.02222222222 + -0.6666666667t + t2 = 0 Move the constant term to the right: Add '-0.02222222222' to each side of the equation. 0.02222222222 + -0.6666666667t + -0.02222222222 + t2 = 0 + -0.02222222222 Reorder the terms: 0.02222222222 + -0.02222222222 + -0.6666666667t + t2 = 0 + -0.02222222222 Combine like terms: 0.02222222222 + -0.02222222222 = 0.00000000000 0.00000000000 + -0.6666666667t + t2 = 0 + -0.02222222222 -0.6666666667t + t2 = 0 + -0.02222222222 Combine like terms: 0 + -0.02222222222 = -0.02222222222 -0.6666666667t + t2 = -0.02222222222 The t term is -0.6666666667t. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667t + 0.1111111112 + t2 = -0.02222222222 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667t + t2 = -0.02222222222 + 0.1111111112 Combine like terms: -0.02222222222 + 0.1111111112 = 0.08888888898 0.1111111112 + -0.6666666667t + t2 = 0.08888888898 Factor a perfect square on the left side: (t + -0.3333333334)(t + -0.3333333334) = 0.08888888898 Calculate the square root of the right side: 0.298142397 Break this problem into two subproblems by setting (t + -0.3333333334) equal to 0.298142397 and -0.298142397.Subproblem 1
t + -0.3333333334 = 0.298142397 Simplifying t + -0.3333333334 = 0.298142397 Reorder the terms: -0.3333333334 + t = 0.298142397 Solving -0.3333333334 + t = 0.298142397 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + t = 0.298142397 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + t = 0.298142397 + 0.3333333334 t = 0.298142397 + 0.3333333334 Combine like terms: 0.298142397 + 0.3333333334 = 0.6314757304 t = 0.6314757304 Simplifying t = 0.6314757304Subproblem 2
t + -0.3333333334 = -0.298142397 Simplifying t + -0.3333333334 = -0.298142397 Reorder the terms: -0.3333333334 + t = -0.298142397 Solving -0.3333333334 + t = -0.298142397 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + t = -0.298142397 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + t = -0.298142397 + 0.3333333334 t = -0.298142397 + 0.3333333334 Combine like terms: -0.298142397 + 0.3333333334 = 0.0351909364 t = 0.0351909364 Simplifying t = 0.0351909364Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.6314757304, 0.0351909364}
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