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Simplifying 0 = -16t2 + 18t + 12 Reorder the terms: 0 = 12 + 18t + -16t2 Solving 0 = 12 + 18t + -16t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -18t + 16t2 = 12 + 18t + -16t2 + -12 + -18t + 16t2 Reorder the terms: -12 + -18t + 16t2 = 12 + -12 + 18t + -18t + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -12 + -18t + 16t2 = 0 + 18t + -18t + -16t2 + 16t2 -12 + -18t + 16t2 = 18t + -18t + -16t2 + 16t2 Combine like terms: 18t + -18t = 0 -12 + -18t + 16t2 = 0 + -16t2 + 16t2 -12 + -18t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12 + -18t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-6 + -9t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-6 + -9t + 8t2)' equal to zero and attempt to solve: Simplifying -6 + -9t + 8t2 = 0 Solving -6 + -9t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.75 + -1.125t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -1.125t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -1.125t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -1.125t + t2 = 0 + 0.75 -1.125t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -1.125t + t2 = 0.75 The t term is -1.125t. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125t + 0.31640625 + t2 = 0.75 + 0.31640625 Reorder the terms: 0.31640625 + -1.125t + t2 = 0.75 + 0.31640625 Combine like terms: 0.75 + 0.31640625 = 1.06640625 0.31640625 + -1.125t + t2 = 1.06640625 Factor a perfect square on the left side: (t + -0.5625)(t + -0.5625) = 1.06640625 Calculate the square root of the right side: 1.032669478 Break this problem into two subproblems by setting (t + -0.5625) equal to 1.032669478 and -1.032669478.Subproblem 1
t + -0.5625 = 1.032669478 Simplifying t + -0.5625 = 1.032669478 Reorder the terms: -0.5625 + t = 1.032669478 Solving -0.5625 + t = 1.032669478 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + t = 1.032669478 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + t = 1.032669478 + 0.5625 t = 1.032669478 + 0.5625 Combine like terms: 1.032669478 + 0.5625 = 1.595169478 t = 1.595169478 Simplifying t = 1.595169478Subproblem 2
t + -0.5625 = -1.032669478 Simplifying t + -0.5625 = -1.032669478 Reorder the terms: -0.5625 + t = -1.032669478 Solving -0.5625 + t = -1.032669478 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + t = -1.032669478 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + t = -1.032669478 + 0.5625 t = -1.032669478 + 0.5625 Combine like terms: -1.032669478 + 0.5625 = -0.470169478 t = -0.470169478 Simplifying t = -0.470169478Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.595169478, -0.470169478}Solution
t = {1.595169478, -0.470169478}
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