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Simplifying 3t(3t) + 5t + -9 = 0 Remove parenthesis around (3t) 3t * 3t + 5t + -9 = 0 Reorder the terms for easier multiplication: 3 * 3t * t + 5t + -9 = 0 Multiply 3 * 3 9t * t + 5t + -9 = 0 Multiply t * t 9t2 + 5t + -9 = 0 Reorder the terms: -9 + 5t + 9t2 = 0 Solving -9 + 5t + 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 + 0.5555555556t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.5555555556t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + 0.5555555556t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.5555555556t + t2 = 0 + 1 0.5555555556t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.5555555556t + t2 = 1 The t term is 0.5555555556t. Take half its coefficient (0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. 0.5555555556t + 0.07716049384 + t2 = 1 + 0.07716049384 Reorder the terms: 0.07716049384 + 0.5555555556t + t2 = 1 + 0.07716049384 Combine like terms: 1 + 0.07716049384 = 1.07716049384 0.07716049384 + 0.5555555556t + t2 = 1.07716049384 Factor a perfect square on the left side: (t + 0.2777777778)(t + 0.2777777778) = 1.07716049384 Calculate the square root of the right side: 1.037863427 Break this problem into two subproblems by setting (t + 0.2777777778) equal to 1.037863427 and -1.037863427.Subproblem 1
t + 0.2777777778 = 1.037863427 Simplifying t + 0.2777777778 = 1.037863427 Reorder the terms: 0.2777777778 + t = 1.037863427 Solving 0.2777777778 + t = 1.037863427 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + t = 1.037863427 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + t = 1.037863427 + -0.2777777778 t = 1.037863427 + -0.2777777778 Combine like terms: 1.037863427 + -0.2777777778 = 0.7600856492 t = 0.7600856492 Simplifying t = 0.7600856492Subproblem 2
t + 0.2777777778 = -1.037863427 Simplifying t + 0.2777777778 = -1.037863427 Reorder the terms: 0.2777777778 + t = -1.037863427 Solving 0.2777777778 + t = -1.037863427 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + t = -1.037863427 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + t = -1.037863427 + -0.2777777778 t = -1.037863427 + -0.2777777778 Combine like terms: -1.037863427 + -0.2777777778 = -1.3156412048 t = -1.3156412048 Simplifying t = -1.3156412048Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.7600856492, -1.3156412048}
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