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Simplifying 45 + 25t + 3t2 = 0 Solving 45 + 25t + 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'. 15 + 8.333333333t + t2 = 0 Move the constant term to the right: Add '-15' to each side of the equation. 15 + 8.333333333t + -15 + t2 = 0 + -15 Reorder the terms: 15 + -15 + 8.333333333t + t2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + 8.333333333t + t2 = 0 + -15 8.333333333t + t2 = 0 + -15 Combine like terms: 0 + -15 = -15 8.333333333t + t2 = -15 The t term is 8.333333333t. Take half its coefficient (4.166666667). Square it (17.36111111) and add it to both sides. Add '17.36111111' to each side of the equation. 8.333333333t + 17.36111111 + t2 = -15 + 17.36111111 Reorder the terms: 17.36111111 + 8.333333333t + t2 = -15 + 17.36111111 Combine like terms: -15 + 17.36111111 = 2.36111111 17.36111111 + 8.333333333t + t2 = 2.36111111 Factor a perfect square on the left side: (t + 4.166666667)(t + 4.166666667) = 2.36111111 Calculate the square root of the right side: 1.536590743 Break this problem into two subproblems by setting (t + 4.166666667) equal to 1.536590743 and -1.536590743.Subproblem 1
t + 4.166666667 = 1.536590743 Simplifying t + 4.166666667 = 1.536590743 Reorder the terms: 4.166666667 + t = 1.536590743 Solving 4.166666667 + t = 1.536590743 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4.166666667' to each side of the equation. 4.166666667 + -4.166666667 + t = 1.536590743 + -4.166666667 Combine like terms: 4.166666667 + -4.166666667 = 0.000000000 0.000000000 + t = 1.536590743 + -4.166666667 t = 1.536590743 + -4.166666667 Combine like terms: 1.536590743 + -4.166666667 = -2.630075924 t = -2.630075924 Simplifying t = -2.630075924Subproblem 2
t + 4.166666667 = -1.536590743 Simplifying t + 4.166666667 = -1.536590743 Reorder the terms: 4.166666667 + t = -1.536590743 Solving 4.166666667 + t = -1.536590743 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4.166666667' to each side of the equation. 4.166666667 + -4.166666667 + t = -1.536590743 + -4.166666667 Combine like terms: 4.166666667 + -4.166666667 = 0.000000000 0.000000000 + t = -1.536590743 + -4.166666667 t = -1.536590743 + -4.166666667 Combine like terms: -1.536590743 + -4.166666667 = -5.70325741 t = -5.70325741 Simplifying t = -5.70325741Solution
The solution to the problem is based on the solutions from the subproblems. t = {-2.630075924, -5.70325741}
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