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Simplifying (4t + 16)(5t + -21) = 180 Reorder the terms: (16 + 4t)(5t + -21) = 180 Reorder the terms: (16 + 4t)(-21 + 5t) = 180 Multiply (16 + 4t) * (-21 + 5t) (16(-21 + 5t) + 4t * (-21 + 5t)) = 180 ((-21 * 16 + 5t * 16) + 4t * (-21 + 5t)) = 180 ((-336 + 80t) + 4t * (-21 + 5t)) = 180 (-336 + 80t + (-21 * 4t + 5t * 4t)) = 180 (-336 + 80t + (-84t + 20t2)) = 180 Combine like terms: 80t + -84t = -4t (-336 + -4t + 20t2) = 180 Solving -336 + -4t + 20t2 = 180 Solving for variable 't'. Reorder the terms: -336 + -180 + -4t + 20t2 = 180 + -180 Combine like terms: -336 + -180 = -516 -516 + -4t + 20t2 = 180 + -180 Combine like terms: 180 + -180 = 0 -516 + -4t + 20t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-129 + -1t + 5t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-129 + -1t + 5t2)' equal to zero and attempt to solve: Simplifying -129 + -1t + 5t2 = 0 Solving -129 + -1t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -25.8 + -0.2t + t2 = 0 Move the constant term to the right: Add '25.8' to each side of the equation. -25.8 + -0.2t + 25.8 + t2 = 0 + 25.8 Reorder the terms: -25.8 + 25.8 + -0.2t + t2 = 0 + 25.8 Combine like terms: -25.8 + 25.8 = 0.0 0.0 + -0.2t + t2 = 0 + 25.8 -0.2t + t2 = 0 + 25.8 Combine like terms: 0 + 25.8 = 25.8 -0.2t + t2 = 25.8 The t term is -0.2t. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2t + 0.01 + t2 = 25.8 + 0.01 Reorder the terms: 0.01 + -0.2t + t2 = 25.8 + 0.01 Combine like terms: 25.8 + 0.01 = 25.81 0.01 + -0.2t + t2 = 25.81 Factor a perfect square on the left side: (t + -0.1)(t + -0.1) = 25.81 Calculate the square root of the right side: 5.080354318 Break this problem into two subproblems by setting (t + -0.1) equal to 5.080354318 and -5.080354318.Subproblem 1
t + -0.1 = 5.080354318 Simplifying t + -0.1 = 5.080354318 Reorder the terms: -0.1 + t = 5.080354318 Solving -0.1 + t = 5.080354318 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + t = 5.080354318 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + t = 5.080354318 + 0.1 t = 5.080354318 + 0.1 Combine like terms: 5.080354318 + 0.1 = 5.180354318 t = 5.180354318 Simplifying t = 5.180354318Subproblem 2
t + -0.1 = -5.080354318 Simplifying t + -0.1 = -5.080354318 Reorder the terms: -0.1 + t = -5.080354318 Solving -0.1 + t = -5.080354318 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + t = -5.080354318 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + t = -5.080354318 + 0.1 t = -5.080354318 + 0.1 Combine like terms: -5.080354318 + 0.1 = -4.980354318 t = -4.980354318 Simplifying t = -4.980354318Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.180354318, -4.980354318}Solution
t = {5.180354318, -4.980354318}
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