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Simplifying 9t2 + -22t + 9 = 0 Reorder the terms: 9 + -22t + 9t2 = 0 Solving 9 + -22t + 9t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 1 + -2.444444444t + t2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.444444444t + -1 + t2 = 0 + -1 Reorder the terms: 1 + -1 + -2.444444444t + t2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.444444444t + t2 = 0 + -1 -2.444444444t + t2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.444444444t + t2 = -1 The t term is -2.444444444t. Take half its coefficient (-1.222222222). Square it (1.493827160) and add it to both sides. Add '1.493827160' to each side of the equation. -2.444444444t + 1.493827160 + t2 = -1 + 1.493827160 Reorder the terms: 1.493827160 + -2.444444444t + t2 = -1 + 1.493827160 Combine like terms: -1 + 1.493827160 = 0.49382716 1.493827160 + -2.444444444t + t2 = 0.49382716 Factor a perfect square on the left side: (t + -1.222222222)(t + -1.222222222) = 0.49382716 Calculate the square root of the right side: 0.702728369 Break this problem into two subproblems by setting (t + -1.222222222) equal to 0.702728369 and -0.702728369.Subproblem 1
t + -1.222222222 = 0.702728369 Simplifying t + -1.222222222 = 0.702728369 Reorder the terms: -1.222222222 + t = 0.702728369 Solving -1.222222222 + t = 0.702728369 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + t = 0.702728369 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + t = 0.702728369 + 1.222222222 t = 0.702728369 + 1.222222222 Combine like terms: 0.702728369 + 1.222222222 = 1.924950591 t = 1.924950591 Simplifying t = 1.924950591Subproblem 2
t + -1.222222222 = -0.702728369 Simplifying t + -1.222222222 = -0.702728369 Reorder the terms: -1.222222222 + t = -0.702728369 Solving -1.222222222 + t = -0.702728369 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + t = -0.702728369 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + t = -0.702728369 + 1.222222222 t = -0.702728369 + 1.222222222 Combine like terms: -0.702728369 + 1.222222222 = 0.519493853 t = 0.519493853 Simplifying t = 0.519493853Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.924950591, 0.519493853}
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