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Simplifying 5t(7 + -1t) + -4(2t + 5) = 0 (7 * 5t + -1t * 5t) + -4(2t + 5) = 0 (35t + -5t2) + -4(2t + 5) = 0 Reorder the terms: 35t + -5t2 + -4(5 + 2t) = 0 35t + -5t2 + (5 * -4 + 2t * -4) = 0 35t + -5t2 + (-20 + -8t) = 0 Reorder the terms: -20 + 35t + -8t + -5t2 = 0 Combine like terms: 35t + -8t = 27t -20 + 27t + -5t2 = 0 Solving -20 + 27t + -5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 4 + -5.4t + t2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -5.4t + -4 + t2 = 0 + -4 Reorder the terms: 4 + -4 + -5.4t + t2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -5.4t + t2 = 0 + -4 -5.4t + t2 = 0 + -4 Combine like terms: 0 + -4 = -4 -5.4t + t2 = -4 The t term is -5.4t. Take half its coefficient (-2.7). Square it (7.29) and add it to both sides. Add '7.29' to each side of the equation. -5.4t + 7.29 + t2 = -4 + 7.29 Reorder the terms: 7.29 + -5.4t + t2 = -4 + 7.29 Combine like terms: -4 + 7.29 = 3.29 7.29 + -5.4t + t2 = 3.29 Factor a perfect square on the left side: (t + -2.7)(t + -2.7) = 3.29 Calculate the square root of the right side: 1.813835715 Break this problem into two subproblems by setting (t + -2.7) equal to 1.813835715 and -1.813835715.Subproblem 1
t + -2.7 = 1.813835715 Simplifying t + -2.7 = 1.813835715 Reorder the terms: -2.7 + t = 1.813835715 Solving -2.7 + t = 1.813835715 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.7' to each side of the equation. -2.7 + 2.7 + t = 1.813835715 + 2.7 Combine like terms: -2.7 + 2.7 = 0.0 0.0 + t = 1.813835715 + 2.7 t = 1.813835715 + 2.7 Combine like terms: 1.813835715 + 2.7 = 4.513835715 t = 4.513835715 Simplifying t = 4.513835715Subproblem 2
t + -2.7 = -1.813835715 Simplifying t + -2.7 = -1.813835715 Reorder the terms: -2.7 + t = -1.813835715 Solving -2.7 + t = -1.813835715 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.7' to each side of the equation. -2.7 + 2.7 + t = -1.813835715 + 2.7 Combine like terms: -2.7 + 2.7 = 0.0 0.0 + t = -1.813835715 + 2.7 t = -1.813835715 + 2.7 Combine like terms: -1.813835715 + 2.7 = 0.886164285 t = 0.886164285 Simplifying t = 0.886164285Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.513835715, 0.886164285}
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