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Simplifying (3t + -15)(2t + -10) = 180 Reorder the terms: (-15 + 3t)(2t + -10) = 180 Reorder the terms: (-15 + 3t)(-10 + 2t) = 180 Multiply (-15 + 3t) * (-10 + 2t) (-15(-10 + 2t) + 3t * (-10 + 2t)) = 180 ((-10 * -15 + 2t * -15) + 3t * (-10 + 2t)) = 180 ((150 + -30t) + 3t * (-10 + 2t)) = 180 (150 + -30t + (-10 * 3t + 2t * 3t)) = 180 (150 + -30t + (-30t + 6t2)) = 180 Combine like terms: -30t + -30t = -60t (150 + -60t + 6t2) = 180 Solving 150 + -60t + 6t2 = 180 Solving for variable 't'. Reorder the terms: 150 + -180 + -60t + 6t2 = 180 + -180 Combine like terms: 150 + -180 = -30 -30 + -60t + 6t2 = 180 + -180 Combine like terms: 180 + -180 = 0 -30 + -60t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-5 + -10t + t2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-5 + -10t + t2)' equal to zero and attempt to solve: Simplifying -5 + -10t + t2 = 0 Solving -5 + -10t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + -10t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -10t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -10t + t2 = 0 + 5 -10t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -10t + t2 = 5 The t term is -10t. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10t + 25 + t2 = 5 + 25 Reorder the terms: 25 + -10t + t2 = 5 + 25 Combine like terms: 5 + 25 = 30 25 + -10t + t2 = 30 Factor a perfect square on the left side: (t + -5)(t + -5) = 30 Calculate the square root of the right side: 5.477225575 Break this problem into two subproblems by setting (t + -5) equal to 5.477225575 and -5.477225575.Subproblem 1
t + -5 = 5.477225575 Simplifying t + -5 = 5.477225575 Reorder the terms: -5 + t = 5.477225575 Solving -5 + t = 5.477225575 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = 5.477225575 + 5 Combine like terms: -5 + 5 = 0 0 + t = 5.477225575 + 5 t = 5.477225575 + 5 Combine like terms: 5.477225575 + 5 = 10.477225575 t = 10.477225575 Simplifying t = 10.477225575Subproblem 2
t + -5 = -5.477225575 Simplifying t + -5 = -5.477225575 Reorder the terms: -5 + t = -5.477225575 Solving -5 + t = -5.477225575 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = -5.477225575 + 5 Combine like terms: -5 + 5 = 0 0 + t = -5.477225575 + 5 t = -5.477225575 + 5 Combine like terms: -5.477225575 + 5 = -0.477225575 t = -0.477225575 Simplifying t = -0.477225575Solution
The solution to the problem is based on the solutions from the subproblems. t = {10.477225575, -0.477225575}Solution
t = {10.477225575, -0.477225575}
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