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Simplifying 0 = 19t2 + -130t + -190 Reorder the terms: 0 = -190 + -130t + 19t2 Solving 0 = -190 + -130t + 19t2 Solving for variable 't'. Combine like terms: 0 + 190 = 190 190 + 130t + -19t2 = -190 + -130t + 19t2 + 190 + 130t + -19t2 Reorder the terms: 190 + 130t + -19t2 = -190 + 190 + -130t + 130t + 19t2 + -19t2 Combine like terms: -190 + 190 = 0 190 + 130t + -19t2 = 0 + -130t + 130t + 19t2 + -19t2 190 + 130t + -19t2 = -130t + 130t + 19t2 + -19t2 Combine like terms: -130t + 130t = 0 190 + 130t + -19t2 = 0 + 19t2 + -19t2 190 + 130t + -19t2 = 19t2 + -19t2 Combine like terms: 19t2 + -19t2 = 0 190 + 130t + -19t2 = 0 Begin completing the square. Divide all terms by -19 the coefficient of the squared term: Divide each side by '-19'. -10 + -6.842105263t + t2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -6.842105263t + 10 + t2 = 0 + 10 Reorder the terms: -10 + 10 + -6.842105263t + t2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -6.842105263t + t2 = 0 + 10 -6.842105263t + t2 = 0 + 10 Combine like terms: 0 + 10 = 10 -6.842105263t + t2 = 10 The t term is -6.842105263t. Take half its coefficient (-3.421052632). Square it (11.70360111) and add it to both sides. Add '11.70360111' to each side of the equation. -6.842105263t + 11.70360111 + t2 = 10 + 11.70360111 Reorder the terms: 11.70360111 + -6.842105263t + t2 = 10 + 11.70360111 Combine like terms: 10 + 11.70360111 = 21.70360111 11.70360111 + -6.842105263t + t2 = 21.70360111 Factor a perfect square on the left side: (t + -3.421052632)(t + -3.421052632) = 21.70360111 Calculate the square root of the right side: 4.658712388 Break this problem into two subproblems by setting (t + -3.421052632) equal to 4.658712388 and -4.658712388.Subproblem 1
t + -3.421052632 = 4.658712388 Simplifying t + -3.421052632 = 4.658712388 Reorder the terms: -3.421052632 + t = 4.658712388 Solving -3.421052632 + t = 4.658712388 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.421052632' to each side of the equation. -3.421052632 + 3.421052632 + t = 4.658712388 + 3.421052632 Combine like terms: -3.421052632 + 3.421052632 = 0.000000000 0.000000000 + t = 4.658712388 + 3.421052632 t = 4.658712388 + 3.421052632 Combine like terms: 4.658712388 + 3.421052632 = 8.07976502 t = 8.07976502 Simplifying t = 8.07976502Subproblem 2
t + -3.421052632 = -4.658712388 Simplifying t + -3.421052632 = -4.658712388 Reorder the terms: -3.421052632 + t = -4.658712388 Solving -3.421052632 + t = -4.658712388 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.421052632' to each side of the equation. -3.421052632 + 3.421052632 + t = -4.658712388 + 3.421052632 Combine like terms: -3.421052632 + 3.421052632 = 0.000000000 0.000000000 + t = -4.658712388 + 3.421052632 t = -4.658712388 + 3.421052632 Combine like terms: -4.658712388 + 3.421052632 = -1.237659756 t = -1.237659756 Simplifying t = -1.237659756Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.07976502, -1.237659756}
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