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Simplifying (5t + -13)(3t + 3) = 180 Reorder the terms: (-13 + 5t)(3t + 3) = 180 Reorder the terms: (-13 + 5t)(3 + 3t) = 180 Multiply (-13 + 5t) * (3 + 3t) (-13(3 + 3t) + 5t * (3 + 3t)) = 180 ((3 * -13 + 3t * -13) + 5t * (3 + 3t)) = 180 ((-39 + -39t) + 5t * (3 + 3t)) = 180 (-39 + -39t + (3 * 5t + 3t * 5t)) = 180 (-39 + -39t + (15t + 15t2)) = 180 Combine like terms: -39t + 15t = -24t (-39 + -24t + 15t2) = 180 Solving -39 + -24t + 15t2 = 180 Solving for variable 't'. Reorder the terms: -39 + -180 + -24t + 15t2 = 180 + -180 Combine like terms: -39 + -180 = -219 -219 + -24t + 15t2 = 180 + -180 Combine like terms: 180 + -180 = 0 -219 + -24t + 15t2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-73 + -8t + 5t2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-73 + -8t + 5t2)' equal to zero and attempt to solve: Simplifying -73 + -8t + 5t2 = 0 Solving -73 + -8t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -14.6 + -1.6t + t2 = 0 Move the constant term to the right: Add '14.6' to each side of the equation. -14.6 + -1.6t + 14.6 + t2 = 0 + 14.6 Reorder the terms: -14.6 + 14.6 + -1.6t + t2 = 0 + 14.6 Combine like terms: -14.6 + 14.6 = 0.0 0.0 + -1.6t + t2 = 0 + 14.6 -1.6t + t2 = 0 + 14.6 Combine like terms: 0 + 14.6 = 14.6 -1.6t + t2 = 14.6 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 = 14.6 + 0.64 Reorder the terms: 0.64 + -1.6t + t2 = 14.6 + 0.64 Combine like terms: 14.6 + 0.64 = 15.24 0.64 + -1.6t + t2 = 15.24 Factor a perfect square on the left side: (t + -0.8)(t + -0.8) = 15.24 Calculate the square root of the right side: 3.903844259 Break this problem into two subproblems by setting (t + -0.8) equal to 3.903844259 and -3.903844259.Subproblem 1
t + -0.8 = 3.903844259 Simplifying t + -0.8 = 3.903844259 Reorder the terms: -0.8 + t = 3.903844259 Solving -0.8 + t = 3.903844259 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 = 3.903844259 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + t = 3.903844259 + 0.8 t = 3.903844259 + 0.8 Combine like terms: 3.903844259 + 0.8 = 4.703844259 t = 4.703844259 Simplifying t = 4.703844259Subproblem 2
t + -0.8 = -3.903844259 Simplifying t + -0.8 = -3.903844259 Reorder the terms: -0.8 + t = -3.903844259 Solving -0.8 + t = -3.903844259 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 = -3.903844259 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + t = -3.903844259 + 0.8 t = -3.903844259 + 0.8 Combine like terms: -3.903844259 + 0.8 = -3.103844259 t = -3.103844259 Simplifying t = -3.103844259Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.703844259, -3.103844259}Solution
t = {4.703844259, -3.103844259}
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