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Simplifying 0 = -16t2 + -6t + 144 Reorder the terms: 0 = 144 + -6t + -16t2 Solving 0 = 144 + -6t + -16t2 Solving for variable 't'. Combine like terms: 0 + -144 = -144 -144 + 6t + 16t2 = 144 + -6t + -16t2 + -144 + 6t + 16t2 Reorder the terms: -144 + 6t + 16t2 = 144 + -144 + -6t + 6t + -16t2 + 16t2 Combine like terms: 144 + -144 = 0 -144 + 6t + 16t2 = 0 + -6t + 6t + -16t2 + 16t2 -144 + 6t + 16t2 = -6t + 6t + -16t2 + 16t2 Combine like terms: -6t + 6t = 0 -144 + 6t + 16t2 = 0 + -16t2 + 16t2 -144 + 6t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -144 + 6t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-72 + 3t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-72 + 3t + 8t2)' equal to zero and attempt to solve: Simplifying -72 + 3t + 8t2 = 0 Solving -72 + 3t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -9 + 0.375t + t2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 0.375t + 9 + t2 = 0 + 9 Reorder the terms: -9 + 9 + 0.375t + t2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 0.375t + t2 = 0 + 9 0.375t + t2 = 0 + 9 Combine like terms: 0 + 9 = 9 0.375t + t2 = 9 The t term is 0.375t. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375t + 0.03515625 + t2 = 9 + 0.03515625 Reorder the terms: 0.03515625 + 0.375t + t2 = 9 + 0.03515625 Combine like terms: 9 + 0.03515625 = 9.03515625 0.03515625 + 0.375t + t2 = 9.03515625 Factor a perfect square on the left side: (t + 0.1875)(t + 0.1875) = 9.03515625 Calculate the square root of the right side: 3.005853664 Break this problem into two subproblems by setting (t + 0.1875) equal to 3.005853664 and -3.005853664.Subproblem 1
t + 0.1875 = 3.005853664 Simplifying t + 0.1875 = 3.005853664 Reorder the terms: 0.1875 + t = 3.005853664 Solving 0.1875 + t = 3.005853664 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = 3.005853664 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = 3.005853664 + -0.1875 t = 3.005853664 + -0.1875 Combine like terms: 3.005853664 + -0.1875 = 2.818353664 t = 2.818353664 Simplifying t = 2.818353664Subproblem 2
t + 0.1875 = -3.005853664 Simplifying t + 0.1875 = -3.005853664 Reorder the terms: 0.1875 + t = -3.005853664 Solving 0.1875 + t = -3.005853664 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = -3.005853664 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = -3.005853664 + -0.1875 t = -3.005853664 + -0.1875 Combine like terms: -3.005853664 + -0.1875 = -3.193353664 t = -3.193353664 Simplifying t = -3.193353664Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.818353664, -3.193353664}Solution
t = {2.818353664, -3.193353664}
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