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Simplifying 7t = -16t2 + 49t + 4 Reorder the terms: 7t = 4 + 49t + -16t2 Solving 7t = 4 + 49t + -16t2 Solving for variable 't'. Reorder the terms: -4 + 7t + -49t + 16t2 = 4 + 49t + -16t2 + -4 + -49t + 16t2 Combine like terms: 7t + -49t = -42t -4 + -42t + 16t2 = 4 + 49t + -16t2 + -4 + -49t + 16t2 Reorder the terms: -4 + -42t + 16t2 = 4 + -4 + 49t + -49t + -16t2 + 16t2 Combine like terms: 4 + -4 = 0 -4 + -42t + 16t2 = 0 + 49t + -49t + -16t2 + 16t2 -4 + -42t + 16t2 = 49t + -49t + -16t2 + 16t2 Combine like terms: 49t + -49t = 0 -4 + -42t + 16t2 = 0 + -16t2 + 16t2 -4 + -42t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -4 + -42t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-2 + -21t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-2 + -21t + 8t2)' equal to zero and attempt to solve: Simplifying -2 + -21t + 8t2 = 0 Solving -2 + -21t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.25 + -2.625t + t2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -2.625t + 0.25 + t2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -2.625t + t2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -2.625t + t2 = 0 + 0.25 -2.625t + t2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -2.625t + t2 = 0.25 The t term is -2.625t. Take half its coefficient (-1.3125). Square it (1.72265625) and add it to both sides. Add '1.72265625' to each side of the equation. -2.625t + 1.72265625 + t2 = 0.25 + 1.72265625 Reorder the terms: 1.72265625 + -2.625t + t2 = 0.25 + 1.72265625 Combine like terms: 0.25 + 1.72265625 = 1.97265625 1.72265625 + -2.625t + t2 = 1.97265625 Factor a perfect square on the left side: (t + -1.3125)(t + -1.3125) = 1.97265625 Calculate the square root of the right side: 1.404512816 Break this problem into two subproblems by setting (t + -1.3125) equal to 1.404512816 and -1.404512816.Subproblem 1
t + -1.3125 = 1.404512816 Simplifying t + -1.3125 = 1.404512816 Reorder the terms: -1.3125 + t = 1.404512816 Solving -1.3125 + t = 1.404512816 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + t = 1.404512816 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + t = 1.404512816 + 1.3125 t = 1.404512816 + 1.3125 Combine like terms: 1.404512816 + 1.3125 = 2.717012816 t = 2.717012816 Simplifying t = 2.717012816Subproblem 2
t + -1.3125 = -1.404512816 Simplifying t + -1.3125 = -1.404512816 Reorder the terms: -1.3125 + t = -1.404512816 Solving -1.3125 + t = -1.404512816 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + t = -1.404512816 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + t = -1.404512816 + 1.3125 t = -1.404512816 + 1.3125 Combine like terms: -1.404512816 + 1.3125 = -0.092012816 t = -0.092012816 Simplifying t = -0.092012816Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.717012816, -0.092012816}Solution
t = {2.717012816, -0.092012816}
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