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Simplifying 0 = 16t2 + 135t + 76 Reorder the terms: 0 = 76 + 135t + 16t2 Solving 0 = 76 + 135t + 16t2 Solving for variable 't'. Combine like terms: 0 + -76 = -76 -76 + -135t + -16t2 = 76 + 135t + 16t2 + -76 + -135t + -16t2 Reorder the terms: -76 + -135t + -16t2 = 76 + -76 + 135t + -135t + 16t2 + -16t2 Combine like terms: 76 + -76 = 0 -76 + -135t + -16t2 = 0 + 135t + -135t + 16t2 + -16t2 -76 + -135t + -16t2 = 135t + -135t + 16t2 + -16t2 Combine like terms: 135t + -135t = 0 -76 + -135t + -16t2 = 0 + 16t2 + -16t2 -76 + -135t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -76 + -135t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(76 + 135t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(76 + 135t + 16t2)' equal to zero and attempt to solve: Simplifying 76 + 135t + 16t2 = 0 Solving 76 + 135t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 4.75 + 8.4375t + t2 = 0 Move the constant term to the right: Add '-4.75' to each side of the equation. 4.75 + 8.4375t + -4.75 + t2 = 0 + -4.75 Reorder the terms: 4.75 + -4.75 + 8.4375t + t2 = 0 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + 8.4375t + t2 = 0 + -4.75 8.4375t + t2 = 0 + -4.75 Combine like terms: 0 + -4.75 = -4.75 8.4375t + t2 = -4.75 The t term is 8.4375t. Take half its coefficient (4.21875). Square it (17.79785156) and add it to both sides. Add '17.79785156' to each side of the equation. 8.4375t + 17.79785156 + t2 = -4.75 + 17.79785156 Reorder the terms: 17.79785156 + 8.4375t + t2 = -4.75 + 17.79785156 Combine like terms: -4.75 + 17.79785156 = 13.04785156 17.79785156 + 8.4375t + t2 = 13.04785156 Factor a perfect square on the left side: (t + 4.21875)(t + 4.21875) = 13.04785156 Calculate the square root of the right side: 3.612180998 Break this problem into two subproblems by setting (t + 4.21875) equal to 3.612180998 and -3.612180998.Subproblem 1
t + 4.21875 = 3.612180998 Simplifying t + 4.21875 = 3.612180998 Reorder the terms: 4.21875 + t = 3.612180998 Solving 4.21875 + t = 3.612180998 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4.21875' to each side of the equation. 4.21875 + -4.21875 + t = 3.612180998 + -4.21875 Combine like terms: 4.21875 + -4.21875 = 0.00000 0.00000 + t = 3.612180998 + -4.21875 t = 3.612180998 + -4.21875 Combine like terms: 3.612180998 + -4.21875 = -0.606569002 t = -0.606569002 Simplifying t = -0.606569002Subproblem 2
t + 4.21875 = -3.612180998 Simplifying t + 4.21875 = -3.612180998 Reorder the terms: 4.21875 + t = -3.612180998 Solving 4.21875 + t = -3.612180998 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4.21875' to each side of the equation. 4.21875 + -4.21875 + t = -3.612180998 + -4.21875 Combine like terms: 4.21875 + -4.21875 = 0.00000 0.00000 + t = -3.612180998 + -4.21875 t = -3.612180998 + -4.21875 Combine like terms: -3.612180998 + -4.21875 = -7.830930998 t = -7.830930998 Simplifying t = -7.830930998Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.606569002, -7.830930998}Solution
t = {-0.606569002, -7.830930998}
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