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Simplifying 5t(t + 5) + -1(t + 2) = 0 Reorder the terms: 5t(5 + t) + -1(t + 2) = 0 (5 * 5t + t * 5t) + -1(t + 2) = 0 (25t + 5t2) + -1(t + 2) = 0 Reorder the terms: 25t + 5t2 + -1(2 + t) = 0 25t + 5t2 + (2 * -1 + t * -1) = 0 25t + 5t2 + (-2 + -1t) = 0 Reorder the terms: -2 + 25t + -1t + 5t2 = 0 Combine like terms: 25t + -1t = 24t -2 + 24t + 5t2 = 0 Solving -2 + 24t + 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'. -0.4 + 4.8t + t2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + 4.8t + 0.4 + t2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + 4.8t + t2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + 4.8t + t2 = 0 + 0.4 4.8t + t2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 4.8t + t2 = 0.4 The t term is 4.8t. Take half its coefficient (2.4). Square it (5.76) and add it to both sides. Add '5.76' to each side of the equation. 4.8t + 5.76 + t2 = 0.4 + 5.76 Reorder the terms: 5.76 + 4.8t + t2 = 0.4 + 5.76 Combine like terms: 0.4 + 5.76 = 6.16 5.76 + 4.8t + t2 = 6.16 Factor a perfect square on the left side: (t + 2.4)(t + 2.4) = 6.16 Calculate the square root of the right side: 2.481934729 Break this problem into two subproblems by setting (t + 2.4) equal to 2.481934729 and -2.481934729.Subproblem 1
t + 2.4 = 2.481934729 Simplifying t + 2.4 = 2.481934729 Reorder the terms: 2.4 + t = 2.481934729 Solving 2.4 + t = 2.481934729 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + t = 2.481934729 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + t = 2.481934729 + -2.4 t = 2.481934729 + -2.4 Combine like terms: 2.481934729 + -2.4 = 0.081934729 t = 0.081934729 Simplifying t = 0.081934729Subproblem 2
t + 2.4 = -2.481934729 Simplifying t + 2.4 = -2.481934729 Reorder the terms: 2.4 + t = -2.481934729 Solving 2.4 + t = -2.481934729 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + t = -2.481934729 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + t = -2.481934729 + -2.4 t = -2.481934729 + -2.4 Combine like terms: -2.481934729 + -2.4 = -4.881934729 t = -4.881934729 Simplifying t = -4.881934729Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.081934729, -4.881934729}
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