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Simplifying 7(t + -6) * 7t = 7(2t + 2) + -11 Reorder the terms: 7(-6 + t) * 7t = 7(2t + 2) + -11 Reorder the terms for easier multiplication: 7 * 7t(-6 + t) = 7(2t + 2) + -11 Multiply 7 * 7 49t(-6 + t) = 7(2t + 2) + -11 (-6 * 49t + t * 49t) = 7(2t + 2) + -11 (-294t + 49t2) = 7(2t + 2) + -11 Reorder the terms: -294t + 49t2 = 7(2 + 2t) + -11 -294t + 49t2 = (2 * 7 + 2t * 7) + -11 -294t + 49t2 = (14 + 14t) + -11 Reorder the terms: -294t + 49t2 = 14 + -11 + 14t Combine like terms: 14 + -11 = 3 -294t + 49t2 = 3 + 14t Solving -294t + 49t2 = 3 + 14t Solving for variable 't'. Reorder the terms: -3 + -294t + -14t + 49t2 = 3 + 14t + -3 + -14t Combine like terms: -294t + -14t = -308t -3 + -308t + 49t2 = 3 + 14t + -3 + -14t Reorder the terms: -3 + -308t + 49t2 = 3 + -3 + 14t + -14t Combine like terms: 3 + -3 = 0 -3 + -308t + 49t2 = 0 + 14t + -14t -3 + -308t + 49t2 = 14t + -14t Combine like terms: 14t + -14t = 0 -3 + -308t + 49t2 = 0 Begin completing the square. Divide all terms by 49 the coefficient of the squared term: Divide each side by '49'. -0.0612244898 + -6.285714286t + t2 = 0 Move the constant term to the right: Add '0.0612244898' to each side of the equation. -0.0612244898 + -6.285714286t + 0.0612244898 + t2 = 0 + 0.0612244898 Reorder the terms: -0.0612244898 + 0.0612244898 + -6.285714286t + t2 = 0 + 0.0612244898 Combine like terms: -0.0612244898 + 0.0612244898 = 0.0000000000 0.0000000000 + -6.285714286t + t2 = 0 + 0.0612244898 -6.285714286t + t2 = 0 + 0.0612244898 Combine like terms: 0 + 0.0612244898 = 0.0612244898 -6.285714286t + t2 = 0.0612244898 The t term is -6.285714286t. Take half its coefficient (-3.142857143). Square it (9.877551021) and add it to both sides. Add '9.877551021' to each side of the equation. -6.285714286t + 9.877551021 + t2 = 0.0612244898 + 9.877551021 Reorder the terms: 9.877551021 + -6.285714286t + t2 = 0.0612244898 + 9.877551021 Combine like terms: 0.0612244898 + 9.877551021 = 9.9387755108 9.877551021 + -6.285714286t + t2 = 9.9387755108 Factor a perfect square on the left side: (t + -3.142857143)(t + -3.142857143) = 9.9387755108 Calculate the square root of the right side: 3.152582356 Break this problem into two subproblems by setting (t + -3.142857143) equal to 3.152582356 and -3.152582356.Subproblem 1
t + -3.142857143 = 3.152582356 Simplifying t + -3.142857143 = 3.152582356 Reorder the terms: -3.142857143 + t = 3.152582356 Solving -3.142857143 + t = 3.152582356 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.142857143' to each side of the equation. -3.142857143 + 3.142857143 + t = 3.152582356 + 3.142857143 Combine like terms: -3.142857143 + 3.142857143 = 0.000000000 0.000000000 + t = 3.152582356 + 3.142857143 t = 3.152582356 + 3.142857143 Combine like terms: 3.152582356 + 3.142857143 = 6.295439499 t = 6.295439499 Simplifying t = 6.295439499Subproblem 2
t + -3.142857143 = -3.152582356 Simplifying t + -3.142857143 = -3.152582356 Reorder the terms: -3.142857143 + t = -3.152582356 Solving -3.142857143 + t = -3.152582356 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.142857143' to each side of the equation. -3.142857143 + 3.142857143 + t = -3.152582356 + 3.142857143 Combine like terms: -3.142857143 + 3.142857143 = 0.000000000 0.000000000 + t = -3.152582356 + 3.142857143 t = -3.152582356 + 3.142857143 Combine like terms: -3.152582356 + 3.142857143 = -0.009725213 t = -0.009725213 Simplifying t = -0.009725213Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.295439499, -0.009725213}
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