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Simplifying 28 = -16t2 + 18t + 24 Reorder the terms: 28 = 24 + 18t + -16t2 Solving 28 = 24 + 18t + -16t2 Solving for variable 't'. Combine like terms: 28 + -24 = 4 4 + -18t + 16t2 = 24 + 18t + -16t2 + -24 + -18t + 16t2 Reorder the terms: 4 + -18t + 16t2 = 24 + -24 + 18t + -18t + -16t2 + 16t2 Combine like terms: 24 + -24 = 0 4 + -18t + 16t2 = 0 + 18t + -18t + -16t2 + 16t2 4 + -18t + 16t2 = 18t + -18t + -16t2 + 16t2 Combine like terms: 18t + -18t = 0 4 + -18t + 16t2 = 0 + -16t2 + 16t2 4 + -18t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 4 + -18t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(2 + -9t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(2 + -9t + 8t2)' equal to zero and attempt to solve: Simplifying 2 + -9t + 8t2 = 0 Solving 2 + -9t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.25 + -1.125t + t2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -1.125t + -0.25 + t2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -1.125t + t2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -1.125t + t2 = 0 + -0.25 -1.125t + t2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -1.125t + t2 = -0.25 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 = -0.25 + 0.31640625 Reorder the terms: 0.31640625 + -1.125t + t2 = -0.25 + 0.31640625 Combine like terms: -0.25 + 0.31640625 = 0.06640625 0.31640625 + -1.125t + t2 = 0.06640625 Factor a perfect square on the left side: (t + -0.5625)(t + -0.5625) = 0.06640625 Calculate the square root of the right side: 0.257694102 Break this problem into two subproblems by setting (t + -0.5625) equal to 0.257694102 and -0.257694102.Subproblem 1
t + -0.5625 = 0.257694102 Simplifying t + -0.5625 = 0.257694102 Reorder the terms: -0.5625 + t = 0.257694102 Solving -0.5625 + t = 0.257694102 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 = 0.257694102 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + t = 0.257694102 + 0.5625 t = 0.257694102 + 0.5625 Combine like terms: 0.257694102 + 0.5625 = 0.820194102 t = 0.820194102 Simplifying t = 0.820194102Subproblem 2
t + -0.5625 = -0.257694102 Simplifying t + -0.5625 = -0.257694102 Reorder the terms: -0.5625 + t = -0.257694102 Solving -0.5625 + t = -0.257694102 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 = -0.257694102 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + t = -0.257694102 + 0.5625 t = -0.257694102 + 0.5625 Combine like terms: -0.257694102 + 0.5625 = 0.304805898 t = 0.304805898 Simplifying t = 0.304805898Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.820194102, 0.304805898}Solution
t = {0.820194102, 0.304805898}
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