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Simplifying 9t2 + 8t + 1 = 0 Reorder the terms: 1 + 8t + 9t2 = 0 Solving 1 + 8t + 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'. 0.1111111111 + 0.8888888889t + t2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + 0.8888888889t + -0.1111111111 + t2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + 0.8888888889t + t2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + 0.8888888889t + t2 = 0 + -0.1111111111 0.8888888889t + t2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 0.8888888889t + t2 = -0.1111111111 The t term is 0.8888888889t. Take half its coefficient (0.4444444445). Square it (0.1975308642) and add it to both sides. Add '0.1975308642' to each side of the equation. 0.8888888889t + 0.1975308642 + t2 = -0.1111111111 + 0.1975308642 Reorder the terms: 0.1975308642 + 0.8888888889t + t2 = -0.1111111111 + 0.1975308642 Combine like terms: -0.1111111111 + 0.1975308642 = 0.0864197531 0.1975308642 + 0.8888888889t + t2 = 0.0864197531 Factor a perfect square on the left side: (t + 0.4444444445)(t + 0.4444444445) = 0.0864197531 Calculate the square root of the right side: 0.293972368 Break this problem into two subproblems by setting (t + 0.4444444445) equal to 0.293972368 and -0.293972368.Subproblem 1
t + 0.4444444445 = 0.293972368 Simplifying t + 0.4444444445 = 0.293972368 Reorder the terms: 0.4444444445 + t = 0.293972368 Solving 0.4444444445 + t = 0.293972368 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + t = 0.293972368 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + t = 0.293972368 + -0.4444444445 t = 0.293972368 + -0.4444444445 Combine like terms: 0.293972368 + -0.4444444445 = -0.1504720765 t = -0.1504720765 Simplifying t = -0.1504720765Subproblem 2
t + 0.4444444445 = -0.293972368 Simplifying t + 0.4444444445 = -0.293972368 Reorder the terms: 0.4444444445 + t = -0.293972368 Solving 0.4444444445 + t = -0.293972368 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + t = -0.293972368 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + t = -0.293972368 + -0.4444444445 t = -0.293972368 + -0.4444444445 Combine like terms: -0.293972368 + -0.4444444445 = -0.7384168125 t = -0.7384168125 Simplifying t = -0.7384168125Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.1504720765, -0.7384168125}
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