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Simplifying 0.5t + 0.5t(t + -0.1) = 1.88 Reorder the terms: 0.5t + 0.5t(-0.1 + t) = 1.88 0.5t + (-0.1 * 0.5t + t * 0.5t) = 1.88 0.5t + (-0.05t + 0.5t2) = 1.88 Combine like terms: 0.5t + -0.05t = 0.45t 0.45t + 0.5t2 = 1.88 Solving 0.45t + 0.5t2 = 1.88 Solving for variable 't'. Reorder the terms: -1.88 + 0.45t + 0.5t2 = 1.88 + -1.88 Combine like terms: 1.88 + -1.88 = 0.00 -1.88 + 0.45t + 0.5t2 = 0.00 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -3.76 + 0.9t + t2 = 0.0 Move the constant term to the right: Add '3.76' to each side of the equation. -3.76 + 0.9t + 3.76 + t2 = 0.0 + 3.76 Reorder the terms: -3.76 + 3.76 + 0.9t + t2 = 0.0 + 3.76 Combine like terms: -3.76 + 3.76 = 0.00 0.00 + 0.9t + t2 = 0.0 + 3.76 0.9t + t2 = 0.0 + 3.76 Combine like terms: 0.0 + 3.76 = 3.76 0.9t + t2 = 3.76 The t term is 0.9t. Take half its coefficient (0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. 0.9t + 0.2025 + t2 = 3.76 + 0.2025 Reorder the terms: 0.2025 + 0.9t + t2 = 3.76 + 0.2025 Combine like terms: 3.76 + 0.2025 = 3.9625 0.2025 + 0.9t + t2 = 3.9625 Factor a perfect square on the left side: (t + 0.45)(t + 0.45) = 3.9625 Calculate the square root of the right side: 1.990602924 Break this problem into two subproblems by setting (t + 0.45) equal to 1.990602924 and -1.990602924.Subproblem 1
t + 0.45 = 1.990602924 Simplifying t + 0.45 = 1.990602924 Reorder the terms: 0.45 + t = 1.990602924 Solving 0.45 + t = 1.990602924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + t = 1.990602924 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + t = 1.990602924 + -0.45 t = 1.990602924 + -0.45 Combine like terms: 1.990602924 + -0.45 = 1.540602924 t = 1.540602924 Simplifying t = 1.540602924Subproblem 2
t + 0.45 = -1.990602924 Simplifying t + 0.45 = -1.990602924 Reorder the terms: 0.45 + t = -1.990602924 Solving 0.45 + t = -1.990602924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + t = -1.990602924 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + t = -1.990602924 + -0.45 t = -1.990602924 + -0.45 Combine like terms: -1.990602924 + -0.45 = -2.440602924 t = -2.440602924 Simplifying t = -2.440602924Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.540602924, -2.440602924}
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