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Simplifying 19 = 6 + 33t + -16t2 Solving 19 = 6 + 33t + -16t2 Solving for variable 't'. Combine like terms: 19 + -6 = 13 13 + -33t + 16t2 = 6 + 33t + -16t2 + -6 + -33t + 16t2 Reorder the terms: 13 + -33t + 16t2 = 6 + -6 + 33t + -33t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 13 + -33t + 16t2 = 0 + 33t + -33t + -16t2 + 16t2 13 + -33t + 16t2 = 33t + -33t + -16t2 + 16t2 Combine like terms: 33t + -33t = 0 13 + -33t + 16t2 = 0 + -16t2 + 16t2 13 + -33t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 13 + -33t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.8125 + -2.0625t + t2 = 0 Move the constant term to the right: Add '-0.8125' to each side of the equation. 0.8125 + -2.0625t + -0.8125 + t2 = 0 + -0.8125 Reorder the terms: 0.8125 + -0.8125 + -2.0625t + t2 = 0 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + -2.0625t + t2 = 0 + -0.8125 -2.0625t + t2 = 0 + -0.8125 Combine like terms: 0 + -0.8125 = -0.8125 -2.0625t + t2 = -0.8125 The t term is -2.0625t. Take half its coefficient (-1.03125). Square it (1.063476563) and add it to both sides. Add '1.063476563' to each side of the equation. -2.0625t + 1.063476563 + t2 = -0.8125 + 1.063476563 Reorder the terms: 1.063476563 + -2.0625t + t2 = -0.8125 + 1.063476563 Combine like terms: -0.8125 + 1.063476563 = 0.250976563 1.063476563 + -2.0625t + t2 = 0.250976563 Factor a perfect square on the left side: (t + -1.03125)(t + -1.03125) = 0.250976563 Calculate the square root of the right side: 0.500975611 Break this problem into two subproblems by setting (t + -1.03125) equal to 0.500975611 and -0.500975611.Subproblem 1
t + -1.03125 = 0.500975611 Simplifying t + -1.03125 = 0.500975611 Reorder the terms: -1.03125 + t = 0.500975611 Solving -1.03125 + t = 0.500975611 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + t = 0.500975611 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + t = 0.500975611 + 1.03125 t = 0.500975611 + 1.03125 Combine like terms: 0.500975611 + 1.03125 = 1.532225611 t = 1.532225611 Simplifying t = 1.532225611Subproblem 2
t + -1.03125 = -0.500975611 Simplifying t + -1.03125 = -0.500975611 Reorder the terms: -1.03125 + t = -0.500975611 Solving -1.03125 + t = -0.500975611 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + t = -0.500975611 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + t = -0.500975611 + 1.03125 t = -0.500975611 + 1.03125 Combine like terms: -0.500975611 + 1.03125 = 0.530274389 t = 0.530274389 Simplifying t = 0.530274389Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.532225611, 0.530274389}
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