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Simplifying 0 = -16t2 + -22t + 180 Reorder the terms: 0 = 180 + -22t + -16t2 Solving 0 = 180 + -22t + -16t2 Solving for variable 't'. Combine like terms: 0 + -180 = -180 -180 + 22t + 16t2 = 180 + -22t + -16t2 + -180 + 22t + 16t2 Reorder the terms: -180 + 22t + 16t2 = 180 + -180 + -22t + 22t + -16t2 + 16t2 Combine like terms: 180 + -180 = 0 -180 + 22t + 16t2 = 0 + -22t + 22t + -16t2 + 16t2 -180 + 22t + 16t2 = -22t + 22t + -16t2 + 16t2 Combine like terms: -22t + 22t = 0 -180 + 22t + 16t2 = 0 + -16t2 + 16t2 -180 + 22t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -180 + 22t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-90 + 11t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-90 + 11t + 8t2)' equal to zero and attempt to solve: Simplifying -90 + 11t + 8t2 = 0 Solving -90 + 11t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -11.25 + 1.375t + t2 = 0 Move the constant term to the right: Add '11.25' to each side of the equation. -11.25 + 1.375t + 11.25 + t2 = 0 + 11.25 Reorder the terms: -11.25 + 11.25 + 1.375t + t2 = 0 + 11.25 Combine like terms: -11.25 + 11.25 = 0.00 0.00 + 1.375t + t2 = 0 + 11.25 1.375t + t2 = 0 + 11.25 Combine like terms: 0 + 11.25 = 11.25 1.375t + t2 = 11.25 The t term is 1.375t. Take half its coefficient (0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. 1.375t + 0.47265625 + t2 = 11.25 + 0.47265625 Reorder the terms: 0.47265625 + 1.375t + t2 = 11.25 + 0.47265625 Combine like terms: 11.25 + 0.47265625 = 11.72265625 0.47265625 + 1.375t + t2 = 11.72265625 Factor a perfect square on the left side: (t + 0.6875)(t + 0.6875) = 11.72265625 Calculate the square root of the right side: 3.423836481 Break this problem into two subproblems by setting (t + 0.6875) equal to 3.423836481 and -3.423836481.Subproblem 1
t + 0.6875 = 3.423836481 Simplifying t + 0.6875 = 3.423836481 Reorder the terms: 0.6875 + t = 3.423836481 Solving 0.6875 + t = 3.423836481 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + t = 3.423836481 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + t = 3.423836481 + -0.6875 t = 3.423836481 + -0.6875 Combine like terms: 3.423836481 + -0.6875 = 2.736336481 t = 2.736336481 Simplifying t = 2.736336481Subproblem 2
t + 0.6875 = -3.423836481 Simplifying t + 0.6875 = -3.423836481 Reorder the terms: 0.6875 + t = -3.423836481 Solving 0.6875 + t = -3.423836481 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + t = -3.423836481 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + t = -3.423836481 + -0.6875 t = -3.423836481 + -0.6875 Combine like terms: -3.423836481 + -0.6875 = -4.111336481 t = -4.111336481 Simplifying t = -4.111336481Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.736336481, -4.111336481}Solution
t = {2.736336481, -4.111336481}
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