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Simplifying (4 + 7t)(4 + 7t) = 12 Multiply (4 + 7t) * (4 + 7t) (4(4 + 7t) + 7t * (4 + 7t)) = 12 ((4 * 4 + 7t * 4) + 7t * (4 + 7t)) = 12 ((16 + 28t) + 7t * (4 + 7t)) = 12 (16 + 28t + (4 * 7t + 7t * 7t)) = 12 (16 + 28t + (28t + 49t2)) = 12 Combine like terms: 28t + 28t = 56t (16 + 56t + 49t2) = 12 Solving 16 + 56t + 49t2 = 12 Solving for variable 't'. Reorder the terms: 16 + -12 + 56t + 49t2 = 12 + -12 Combine like terms: 16 + -12 = 4 4 + 56t + 49t2 = 12 + -12 Combine like terms: 12 + -12 = 0 4 + 56t + 49t2 = 0 Begin completing the square. Divide all terms by 49 the coefficient of the squared term: Divide each side by '49'. 0.08163265306 + 1.142857143t + t2 = 0 Move the constant term to the right: Add '-0.08163265306' to each side of the equation. 0.08163265306 + 1.142857143t + -0.08163265306 + t2 = 0 + -0.08163265306 Reorder the terms: 0.08163265306 + -0.08163265306 + 1.142857143t + t2 = 0 + -0.08163265306 Combine like terms: 0.08163265306 + -0.08163265306 = 0.00000000000 0.00000000000 + 1.142857143t + t2 = 0 + -0.08163265306 1.142857143t + t2 = 0 + -0.08163265306 Combine like terms: 0 + -0.08163265306 = -0.08163265306 1.142857143t + t2 = -0.08163265306 The t term is 1.142857143t. Take half its coefficient (0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. 1.142857143t + 0.3265306123 + t2 = -0.08163265306 + 0.3265306123 Reorder the terms: 0.3265306123 + 1.142857143t + t2 = -0.08163265306 + 0.3265306123 Combine like terms: -0.08163265306 + 0.3265306123 = 0.24489795924 0.3265306123 + 1.142857143t + t2 = 0.24489795924 Factor a perfect square on the left side: (t + 0.5714285715)(t + 0.5714285715) = 0.24489795924 Calculate the square root of the right side: 0.494871659 Break this problem into two subproblems by setting (t + 0.5714285715) equal to 0.494871659 and -0.494871659.Subproblem 1
t + 0.5714285715 = 0.494871659 Simplifying t + 0.5714285715 = 0.494871659 Reorder the terms: 0.5714285715 + t = 0.494871659 Solving 0.5714285715 + t = 0.494871659 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + t = 0.494871659 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + t = 0.494871659 + -0.5714285715 t = 0.494871659 + -0.5714285715 Combine like terms: 0.494871659 + -0.5714285715 = -0.0765569125 t = -0.0765569125 Simplifying t = -0.0765569125Subproblem 2
t + 0.5714285715 = -0.494871659 Simplifying t + 0.5714285715 = -0.494871659 Reorder the terms: 0.5714285715 + t = -0.494871659 Solving 0.5714285715 + t = -0.494871659 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + t = -0.494871659 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + t = -0.494871659 + -0.5714285715 t = -0.494871659 + -0.5714285715 Combine like terms: -0.494871659 + -0.5714285715 = -1.0663002305 t = -1.0663002305 Simplifying t = -1.0663002305Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.0765569125, -1.0663002305}
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