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Simplifying 9t2 + 32t + 24 = 0 Reorder the terms: 24 + 32t + 9t2 = 0 Solving 24 + 32t + 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'. 2.666666667 + 3.555555556t + t2 = 0 Move the constant term to the right: Add '-2.666666667' to each side of the equation. 2.666666667 + 3.555555556t + -2.666666667 + t2 = 0 + -2.666666667 Reorder the terms: 2.666666667 + -2.666666667 + 3.555555556t + t2 = 0 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + 3.555555556t + t2 = 0 + -2.666666667 3.555555556t + t2 = 0 + -2.666666667 Combine like terms: 0 + -2.666666667 = -2.666666667 3.555555556t + t2 = -2.666666667 The t term is 3.555555556t. Take half its coefficient (1.777777778). Square it (3.160493828) and add it to both sides. Add '3.160493828' to each side of the equation. 3.555555556t + 3.160493828 + t2 = -2.666666667 + 3.160493828 Reorder the terms: 3.160493828 + 3.555555556t + t2 = -2.666666667 + 3.160493828 Combine like terms: -2.666666667 + 3.160493828 = 0.493827161 3.160493828 + 3.555555556t + t2 = 0.493827161 Factor a perfect square on the left side: (t + 1.777777778)(t + 1.777777778) = 0.493827161 Calculate the square root of the right side: 0.702728369 Break this problem into two subproblems by setting (t + 1.777777778) equal to 0.702728369 and -0.702728369.Subproblem 1
t + 1.777777778 = 0.702728369 Simplifying t + 1.777777778 = 0.702728369 Reorder the terms: 1.777777778 + t = 0.702728369 Solving 1.777777778 + t = 0.702728369 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.777777778' to each side of the equation. 1.777777778 + -1.777777778 + t = 0.702728369 + -1.777777778 Combine like terms: 1.777777778 + -1.777777778 = 0.000000000 0.000000000 + t = 0.702728369 + -1.777777778 t = 0.702728369 + -1.777777778 Combine like terms: 0.702728369 + -1.777777778 = -1.075049409 t = -1.075049409 Simplifying t = -1.075049409Subproblem 2
t + 1.777777778 = -0.702728369 Simplifying t + 1.777777778 = -0.702728369 Reorder the terms: 1.777777778 + t = -0.702728369 Solving 1.777777778 + t = -0.702728369 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.777777778' to each side of the equation. 1.777777778 + -1.777777778 + t = -0.702728369 + -1.777777778 Combine like terms: 1.777777778 + -1.777777778 = 0.000000000 0.000000000 + t = -0.702728369 + -1.777777778 t = -0.702728369 + -1.777777778 Combine like terms: -0.702728369 + -1.777777778 = -2.480506147 t = -2.480506147 Simplifying t = -2.480506147Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.075049409, -2.480506147}
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