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Simplifying 9t2 + 30t + 11 = 0 Reorder the terms: 11 + 30t + 9t2 = 0 Solving 11 + 30t + 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'. 1.222222222 + 3.333333333t + t2 = 0 Move the constant term to the right: Add '-1.222222222' to each side of the equation. 1.222222222 + 3.333333333t + -1.222222222 + t2 = 0 + -1.222222222 Reorder the terms: 1.222222222 + -1.222222222 + 3.333333333t + t2 = 0 + -1.222222222 Combine like terms: 1.222222222 + -1.222222222 = 0.000000000 0.000000000 + 3.333333333t + t2 = 0 + -1.222222222 3.333333333t + t2 = 0 + -1.222222222 Combine like terms: 0 + -1.222222222 = -1.222222222 3.333333333t + t2 = -1.222222222 The t term is 3.333333333t. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333t + 2.777777779 + t2 = -1.222222222 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333t + t2 = -1.222222222 + 2.777777779 Combine like terms: -1.222222222 + 2.777777779 = 1.555555557 2.777777779 + 3.333333333t + t2 = 1.555555557 Factor a perfect square on the left side: (t + 1.666666667)(t + 1.666666667) = 1.555555557 Calculate the square root of the right side: 1.24721913 Break this problem into two subproblems by setting (t + 1.666666667) equal to 1.24721913 and -1.24721913.Subproblem 1
t + 1.666666667 = 1.24721913 Simplifying t + 1.666666667 = 1.24721913 Reorder the terms: 1.666666667 + t = 1.24721913 Solving 1.666666667 + t = 1.24721913 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + t = 1.24721913 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + t = 1.24721913 + -1.666666667 t = 1.24721913 + -1.666666667 Combine like terms: 1.24721913 + -1.666666667 = -0.419447537 t = -0.419447537 Simplifying t = -0.419447537Subproblem 2
t + 1.666666667 = -1.24721913 Simplifying t + 1.666666667 = -1.24721913 Reorder the terms: 1.666666667 + t = -1.24721913 Solving 1.666666667 + t = -1.24721913 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + t = -1.24721913 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + t = -1.24721913 + -1.666666667 t = -1.24721913 + -1.666666667 Combine like terms: -1.24721913 + -1.666666667 = -2.913885797 t = -2.913885797 Simplifying t = -2.913885797Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.419447537, -2.913885797}
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