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Simplifying 6 = -16t2 + 24t Reorder the terms: 6 = 24t + -16t2 Solving 6 = 24t + -16t2 Solving for variable 't'. Reorder the terms: 6 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 6 + -24t + 16t2 = 0 + -16t2 + 16t2 6 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 6 + -24t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(3 + -12t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(3 + -12t + 8t2)' equal to zero and attempt to solve: Simplifying 3 + -12t + 8t2 = 0 Solving 3 + -12t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.375 + -1.5t + t2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + -1.5t + -0.375 + t2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + -1.5t + t2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + -1.5t + t2 = 0 + -0.375 -1.5t + t2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 -1.5t + t2 = -0.375 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = -0.375 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = -0.375 + 0.5625 Combine like terms: -0.375 + 0.5625 = 0.1875 0.5625 + -1.5t + t2 = 0.1875 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (t + -0.75) equal to 0.433012702 and -0.433012702.Subproblem 1
t + -0.75 = 0.433012702 Simplifying t + -0.75 = 0.433012702 Reorder the terms: -0.75 + t = 0.433012702 Solving -0.75 + t = 0.433012702 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 0.433012702 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.433012702 + 0.75 t = 0.433012702 + 0.75 Combine like terms: 0.433012702 + 0.75 = 1.183012702 t = 1.183012702 Simplifying t = 1.183012702Subproblem 2
t + -0.75 = -0.433012702 Simplifying t + -0.75 = -0.433012702 Reorder the terms: -0.75 + t = -0.433012702 Solving -0.75 + t = -0.433012702 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -0.433012702 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.433012702 + 0.75 t = -0.433012702 + 0.75 Combine like terms: -0.433012702 + 0.75 = 0.316987298 t = 0.316987298 Simplifying t = 0.316987298Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.183012702, 0.316987298}Solution
t = {1.183012702, 0.316987298}
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