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Simplifying 3t2 + -32t + 9 = 0 Reorder the terms: 9 + -32t + 3t2 = 0 Solving 9 + -32t + 3t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3 + -10.66666667t + t2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -10.66666667t + -3 + t2 = 0 + -3 Reorder the terms: 3 + -3 + -10.66666667t + t2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -10.66666667t + t2 = 0 + -3 -10.66666667t + t2 = 0 + -3 Combine like terms: 0 + -3 = -3 -10.66666667t + t2 = -3 The t term is -10.66666667t. Take half its coefficient (-5.333333335). Square it (28.44444446) and add it to both sides. Add '28.44444446' to each side of the equation. -10.66666667t + 28.44444446 + t2 = -3 + 28.44444446 Reorder the terms: 28.44444446 + -10.66666667t + t2 = -3 + 28.44444446 Combine like terms: -3 + 28.44444446 = 25.44444446 28.44444446 + -10.66666667t + t2 = 25.44444446 Factor a perfect square on the left side: (t + -5.333333335)(t + -5.333333335) = 25.44444446 Calculate the square root of the right side: 5.044248652 Break this problem into two subproblems by setting (t + -5.333333335) equal to 5.044248652 and -5.044248652.Subproblem 1
t + -5.333333335 = 5.044248652 Simplifying t + -5.333333335 = 5.044248652 Reorder the terms: -5.333333335 + t = 5.044248652 Solving -5.333333335 + t = 5.044248652 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + t = 5.044248652 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + t = 5.044248652 + 5.333333335 t = 5.044248652 + 5.333333335 Combine like terms: 5.044248652 + 5.333333335 = 10.377581987 t = 10.377581987 Simplifying t = 10.377581987Subproblem 2
t + -5.333333335 = -5.044248652 Simplifying t + -5.333333335 = -5.044248652 Reorder the terms: -5.333333335 + t = -5.044248652 Solving -5.333333335 + t = -5.044248652 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + t = -5.044248652 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + t = -5.044248652 + 5.333333335 t = -5.044248652 + 5.333333335 Combine like terms: -5.044248652 + 5.333333335 = 0.289084683 t = 0.289084683 Simplifying t = 0.289084683Solution
The solution to the problem is based on the solutions from the subproblems. t = {10.377581987, 0.289084683}
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