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Simplifying 3t + 1.6 = 5t2 + 11t + 1.4 Reorder the terms: 1.6 + 3t = 5t2 + 11t + 1.4 Reorder the terms: 1.6 + 3t = 1.4 + 11t + 5t2 Solving 1.6 + 3t = 1.4 + 11t + 5t2 Solving for variable 't'. Reorder the terms: 1.6 + -1.4 + 3t + -11t + -5t2 = 1.4 + 11t + 5t2 + -1.4 + -11t + -5t2 Combine like terms: 1.6 + -1.4 = 0.2 0.2 + 3t + -11t + -5t2 = 1.4 + 11t + 5t2 + -1.4 + -11t + -5t2 Combine like terms: 3t + -11t = -8t 0.2 + -8t + -5t2 = 1.4 + 11t + 5t2 + -1.4 + -11t + -5t2 Reorder the terms: 0.2 + -8t + -5t2 = 1.4 + -1.4 + 11t + -11t + 5t2 + -5t2 Combine like terms: 1.4 + -1.4 = 0.0 0.2 + -8t + -5t2 = 0.0 + 11t + -11t + 5t2 + -5t2 0.2 + -8t + -5t2 = 11t + -11t + 5t2 + -5t2 Combine like terms: 11t + -11t = 0 0.2 + -8t + -5t2 = 0 + 5t2 + -5t2 0.2 + -8t + -5t2 = 5t2 + -5t2 Combine like terms: 5t2 + -5t2 = 0 0.2 + -8t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -0.04 + 1.6t + t2 = 0 Move the constant term to the right: Add '0.04' to each side of the equation. -0.04 + 1.6t + 0.04 + t2 = 0 + 0.04 Reorder the terms: -0.04 + 0.04 + 1.6t + t2 = 0 + 0.04 Combine like terms: -0.04 + 0.04 = 0.00 0.00 + 1.6t + t2 = 0 + 0.04 1.6t + t2 = 0 + 0.04 Combine like terms: 0 + 0.04 = 0.04 1.6t + t2 = 0.04 The t term is 1.6t. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6t + 0.64 + t2 = 0.04 + 0.64 Reorder the terms: 0.64 + 1.6t + t2 = 0.04 + 0.64 Combine like terms: 0.04 + 0.64 = 0.68 0.64 + 1.6t + t2 = 0.68 Factor a perfect square on the left side: (t + 0.8)(t + 0.8) = 0.68 Calculate the square root of the right side: 0.824621125 Break this problem into two subproblems by setting (t + 0.8) equal to 0.824621125 and -0.824621125.Subproblem 1
t + 0.8 = 0.824621125 Simplifying t + 0.8 = 0.824621125 Reorder the terms: 0.8 + t = 0.824621125 Solving 0.8 + t = 0.824621125 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + t = 0.824621125 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + t = 0.824621125 + -0.8 t = 0.824621125 + -0.8 Combine like terms: 0.824621125 + -0.8 = 0.024621125 t = 0.024621125 Simplifying t = 0.024621125Subproblem 2
t + 0.8 = -0.824621125 Simplifying t + 0.8 = -0.824621125 Reorder the terms: 0.8 + t = -0.824621125 Solving 0.8 + t = -0.824621125 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + t = -0.824621125 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + t = -0.824621125 + -0.8 t = -0.824621125 + -0.8 Combine like terms: -0.824621125 + -0.8 = -1.624621125 t = -1.624621125 Simplifying t = -1.624621125Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.024621125, -1.624621125}
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