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Simplifying -4(3t + -2) * 2t = 4t + -5 Reorder the terms: -4(-2 + 3t) * 2t = 4t + -5 Reorder the terms for easier multiplication: -4 * 2t(-2 + 3t) = 4t + -5 Multiply -4 * 2 -8t(-2 + 3t) = 4t + -5 (-2 * -8t + 3t * -8t) = 4t + -5 (16t + -24t2) = 4t + -5 Reorder the terms: 16t + -24t2 = -5 + 4t Solving 16t + -24t2 = -5 + 4t Solving for variable 't'. Reorder the terms: 5 + 16t + -4t + -24t2 = -5 + 4t + 5 + -4t Combine like terms: 16t + -4t = 12t 5 + 12t + -24t2 = -5 + 4t + 5 + -4t Reorder the terms: 5 + 12t + -24t2 = -5 + 5 + 4t + -4t Combine like terms: -5 + 5 = 0 5 + 12t + -24t2 = 0 + 4t + -4t 5 + 12t + -24t2 = 4t + -4t Combine like terms: 4t + -4t = 0 5 + 12t + -24t2 = 0 Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -0.2083333333 + -0.5t + t2 = 0 Move the constant term to the right: Add '0.2083333333' to each side of the equation. -0.2083333333 + -0.5t + 0.2083333333 + t2 = 0 + 0.2083333333 Reorder the terms: -0.2083333333 + 0.2083333333 + -0.5t + t2 = 0 + 0.2083333333 Combine like terms: -0.2083333333 + 0.2083333333 = 0.0000000000 0.0000000000 + -0.5t + t2 = 0 + 0.2083333333 -0.5t + t2 = 0 + 0.2083333333 Combine like terms: 0 + 0.2083333333 = 0.2083333333 -0.5t + t2 = 0.2083333333 The t term is -0.5t. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5t + 0.0625 + t2 = 0.2083333333 + 0.0625 Reorder the terms: 0.0625 + -0.5t + t2 = 0.2083333333 + 0.0625 Combine like terms: 0.2083333333 + 0.0625 = 0.2708333333 0.0625 + -0.5t + t2 = 0.2708333333 Factor a perfect square on the left side: (t + -0.25)(t + -0.25) = 0.2708333333 Calculate the square root of the right side: 0.5204165 Break this problem into two subproblems by setting (t + -0.25) equal to 0.5204165 and -0.5204165.Subproblem 1
t + -0.25 = 0.5204165 Simplifying t + -0.25 = 0.5204165 Reorder the terms: -0.25 + t = 0.5204165 Solving -0.25 + t = 0.5204165 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + t = 0.5204165 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + t = 0.5204165 + 0.25 t = 0.5204165 + 0.25 Combine like terms: 0.5204165 + 0.25 = 0.7704165 t = 0.7704165 Simplifying t = 0.7704165Subproblem 2
t + -0.25 = -0.5204165 Simplifying t + -0.25 = -0.5204165 Reorder the terms: -0.25 + t = -0.5204165 Solving -0.25 + t = -0.5204165 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + t = -0.5204165 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + t = -0.5204165 + 0.25 t = -0.5204165 + 0.25 Combine like terms: -0.5204165 + 0.25 = -0.2704165 t = -0.2704165 Simplifying t = -0.2704165Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.7704165, -0.2704165}
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