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Simplifying 16t2 + -16t + 4 = 30t + -13 Reorder the terms: 4 + -16t + 16t2 = 30t + -13 Reorder the terms: 4 + -16t + 16t2 = -13 + 30t Solving 4 + -16t + 16t2 = -13 + 30t Solving for variable 't'. Reorder the terms: 4 + 13 + -16t + -30t + 16t2 = -13 + 30t + 13 + -30t Combine like terms: 4 + 13 = 17 17 + -16t + -30t + 16t2 = -13 + 30t + 13 + -30t Combine like terms: -16t + -30t = -46t 17 + -46t + 16t2 = -13 + 30t + 13 + -30t Reorder the terms: 17 + -46t + 16t2 = -13 + 13 + 30t + -30t Combine like terms: -13 + 13 = 0 17 + -46t + 16t2 = 0 + 30t + -30t 17 + -46t + 16t2 = 30t + -30t Combine like terms: 30t + -30t = 0 17 + -46t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.0625 + -2.875t + t2 = 0 Move the constant term to the right: Add '-1.0625' to each side of the equation. 1.0625 + -2.875t + -1.0625 + t2 = 0 + -1.0625 Reorder the terms: 1.0625 + -1.0625 + -2.875t + t2 = 0 + -1.0625 Combine like terms: 1.0625 + -1.0625 = 0.0000 0.0000 + -2.875t + t2 = 0 + -1.0625 -2.875t + t2 = 0 + -1.0625 Combine like terms: 0 + -1.0625 = -1.0625 -2.875t + t2 = -1.0625 The t term is -2.875t. Take half its coefficient (-1.4375). Square it (2.06640625) and add it to both sides. Add '2.06640625' to each side of the equation. -2.875t + 2.06640625 + t2 = -1.0625 + 2.06640625 Reorder the terms: 2.06640625 + -2.875t + t2 = -1.0625 + 2.06640625 Combine like terms: -1.0625 + 2.06640625 = 1.00390625 2.06640625 + -2.875t + t2 = 1.00390625 Factor a perfect square on the left side: (t + -1.4375)(t + -1.4375) = 1.00390625 Calculate the square root of the right side: 1.001951221 Break this problem into two subproblems by setting (t + -1.4375) equal to 1.001951221 and -1.001951221.Subproblem 1
t + -1.4375 = 1.001951221 Simplifying t + -1.4375 = 1.001951221 Reorder the terms: -1.4375 + t = 1.001951221 Solving -1.4375 + t = 1.001951221 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + t = 1.001951221 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + t = 1.001951221 + 1.4375 t = 1.001951221 + 1.4375 Combine like terms: 1.001951221 + 1.4375 = 2.439451221 t = 2.439451221 Simplifying t = 2.439451221Subproblem 2
t + -1.4375 = -1.001951221 Simplifying t + -1.4375 = -1.001951221 Reorder the terms: -1.4375 + t = -1.001951221 Solving -1.4375 + t = -1.001951221 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + t = -1.001951221 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + t = -1.001951221 + 1.4375 t = -1.001951221 + 1.4375 Combine like terms: -1.001951221 + 1.4375 = 0.435548779 t = 0.435548779 Simplifying t = 0.435548779Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.439451221, 0.435548779}
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