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Simplifying 3t + -1(7 + -11t) = t(5t + -5) 3t + (7 * -1 + -11t * -1) = t(5t + -5) 3t + (-7 + 11t) = t(5t + -5) Reorder the terms: -7 + 3t + 11t = t(5t + -5) Combine like terms: 3t + 11t = 14t -7 + 14t = t(5t + -5) Reorder the terms: -7 + 14t = t(-5 + 5t) -7 + 14t = (-5 * t + 5t * t) -7 + 14t = (-5t + 5t2) Solving -7 + 14t = -5t + 5t2 Solving for variable 't'. Combine like terms: 14t + 5t = 19t -7 + 19t + -5t2 = -5t + 5t2 + 5t + -5t2 Reorder the terms: -7 + 19t + -5t2 = -5t + 5t + 5t2 + -5t2 Combine like terms: -5t + 5t = 0 -7 + 19t + -5t2 = 0 + 5t2 + -5t2 -7 + 19t + -5t2 = 5t2 + -5t2 Combine like terms: 5t2 + -5t2 = 0 -7 + 19t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 1.4 + -3.8t + t2 = 0 Move the constant term to the right: Add '-1.4' to each side of the equation. 1.4 + -3.8t + -1.4 + t2 = 0 + -1.4 Reorder the terms: 1.4 + -1.4 + -3.8t + t2 = 0 + -1.4 Combine like terms: 1.4 + -1.4 = 0.0 0.0 + -3.8t + t2 = 0 + -1.4 -3.8t + t2 = 0 + -1.4 Combine like terms: 0 + -1.4 = -1.4 -3.8t + t2 = -1.4 The t term is -3.8t. Take half its coefficient (-1.9). Square it (3.61) and add it to both sides. Add '3.61' to each side of the equation. -3.8t + 3.61 + t2 = -1.4 + 3.61 Reorder the terms: 3.61 + -3.8t + t2 = -1.4 + 3.61 Combine like terms: -1.4 + 3.61 = 2.21 3.61 + -3.8t + t2 = 2.21 Factor a perfect square on the left side: (t + -1.9)(t + -1.9) = 2.21 Calculate the square root of the right side: 1.486606875 Break this problem into two subproblems by setting (t + -1.9) equal to 1.486606875 and -1.486606875.Subproblem 1
t + -1.9 = 1.486606875 Simplifying t + -1.9 = 1.486606875 Reorder the terms: -1.9 + t = 1.486606875 Solving -1.9 + t = 1.486606875 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.9' to each side of the equation. -1.9 + 1.9 + t = 1.486606875 + 1.9 Combine like terms: -1.9 + 1.9 = 0.0 0.0 + t = 1.486606875 + 1.9 t = 1.486606875 + 1.9 Combine like terms: 1.486606875 + 1.9 = 3.386606875 t = 3.386606875 Simplifying t = 3.386606875Subproblem 2
t + -1.9 = -1.486606875 Simplifying t + -1.9 = -1.486606875 Reorder the terms: -1.9 + t = -1.486606875 Solving -1.9 + t = -1.486606875 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.9' to each side of the equation. -1.9 + 1.9 + t = -1.486606875 + 1.9 Combine like terms: -1.9 + 1.9 = 0.0 0.0 + t = -1.486606875 + 1.9 t = -1.486606875 + 1.9 Combine like terms: -1.486606875 + 1.9 = 0.413393125 t = 0.413393125 Simplifying t = 0.413393125Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.386606875, 0.413393125}
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