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Simplifying 7(t + -5) * 7t = 7(2t + 4) + -11 Reorder the terms: 7(-5 + t) * 7t = 7(2t + 4) + -11 Reorder the terms for easier multiplication: 7 * 7t(-5 + t) = 7(2t + 4) + -11 Multiply 7 * 7 49t(-5 + t) = 7(2t + 4) + -11 (-5 * 49t + t * 49t) = 7(2t + 4) + -11 (-245t + 49t2) = 7(2t + 4) + -11 Reorder the terms: -245t + 49t2 = 7(4 + 2t) + -11 -245t + 49t2 = (4 * 7 + 2t * 7) + -11 -245t + 49t2 = (28 + 14t) + -11 Reorder the terms: -245t + 49t2 = 28 + -11 + 14t Combine like terms: 28 + -11 = 17 -245t + 49t2 = 17 + 14t Solving -245t + 49t2 = 17 + 14t Solving for variable 't'. Reorder the terms: -17 + -245t + -14t + 49t2 = 17 + 14t + -17 + -14t Combine like terms: -245t + -14t = -259t -17 + -259t + 49t2 = 17 + 14t + -17 + -14t Reorder the terms: -17 + -259t + 49t2 = 17 + -17 + 14t + -14t Combine like terms: 17 + -17 = 0 -17 + -259t + 49t2 = 0 + 14t + -14t -17 + -259t + 49t2 = 14t + -14t Combine like terms: 14t + -14t = 0 -17 + -259t + 49t2 = 0 Begin completing the square. Divide all terms by 49 the coefficient of the squared term: Divide each side by '49'. -0.3469387755 + -5.285714286t + t2 = 0 Move the constant term to the right: Add '0.3469387755' to each side of the equation. -0.3469387755 + -5.285714286t + 0.3469387755 + t2 = 0 + 0.3469387755 Reorder the terms: -0.3469387755 + 0.3469387755 + -5.285714286t + t2 = 0 + 0.3469387755 Combine like terms: -0.3469387755 + 0.3469387755 = 0.0000000000 0.0000000000 + -5.285714286t + t2 = 0 + 0.3469387755 -5.285714286t + t2 = 0 + 0.3469387755 Combine like terms: 0 + 0.3469387755 = 0.3469387755 -5.285714286t + t2 = 0.3469387755 The t term is -5.285714286t. Take half its coefficient (-2.642857143). Square it (6.984693878) and add it to both sides. Add '6.984693878' to each side of the equation. -5.285714286t + 6.984693878 + t2 = 0.3469387755 + 6.984693878 Reorder the terms: 6.984693878 + -5.285714286t + t2 = 0.3469387755 + 6.984693878 Combine like terms: 0.3469387755 + 6.984693878 = 7.3316326535 6.984693878 + -5.285714286t + t2 = 7.3316326535 Factor a perfect square on the left side: (t + -2.642857143)(t + -2.642857143) = 7.3316326535 Calculate the square root of the right side: 2.707698775 Break this problem into two subproblems by setting (t + -2.642857143) equal to 2.707698775 and -2.707698775.Subproblem 1
t + -2.642857143 = 2.707698775 Simplifying t + -2.642857143 = 2.707698775 Reorder the terms: -2.642857143 + t = 2.707698775 Solving -2.642857143 + t = 2.707698775 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + t = 2.707698775 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + t = 2.707698775 + 2.642857143 t = 2.707698775 + 2.642857143 Combine like terms: 2.707698775 + 2.642857143 = 5.350555918 t = 5.350555918 Simplifying t = 5.350555918Subproblem 2
t + -2.642857143 = -2.707698775 Simplifying t + -2.642857143 = -2.707698775 Reorder the terms: -2.642857143 + t = -2.707698775 Solving -2.642857143 + t = -2.707698775 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + t = -2.707698775 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + t = -2.707698775 + 2.642857143 t = -2.707698775 + 2.642857143 Combine like terms: -2.707698775 + 2.642857143 = -0.064841632 t = -0.064841632 Simplifying t = -0.064841632Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.350555918, -0.064841632}
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