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Simplifying -16t2 + 18t + 24 = 0 Reorder the terms: 24 + 18t + -16t2 = 0 Solving 24 + 18t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(12 + 9t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(12 + 9t + -8t2)' equal to zero and attempt to solve: Simplifying 12 + 9t + -8t2 = 0 Solving 12 + 9t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.5 + -1.125t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -1.125t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -1.125t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -1.125t + t2 = 0 + 1.5 -1.125t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -1.125t + t2 = 1.5 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 = 1.5 + 0.31640625 Reorder the terms: 0.31640625 + -1.125t + t2 = 1.5 + 0.31640625 Combine like terms: 1.5 + 0.31640625 = 1.81640625 0.31640625 + -1.125t + t2 = 1.81640625 Factor a perfect square on the left side: (t + -0.5625)(t + -0.5625) = 1.81640625 Calculate the square root of the right side: 1.347741166 Break this problem into two subproblems by setting (t + -0.5625) equal to 1.347741166 and -1.347741166.Subproblem 1
t + -0.5625 = 1.347741166 Simplifying t + -0.5625 = 1.347741166 Reorder the terms: -0.5625 + t = 1.347741166 Solving -0.5625 + t = 1.347741166 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 = 1.347741166 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + t = 1.347741166 + 0.5625 t = 1.347741166 + 0.5625 Combine like terms: 1.347741166 + 0.5625 = 1.910241166 t = 1.910241166 Simplifying t = 1.910241166Subproblem 2
t + -0.5625 = -1.347741166 Simplifying t + -0.5625 = -1.347741166 Reorder the terms: -0.5625 + t = -1.347741166 Solving -0.5625 + t = -1.347741166 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 = -1.347741166 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + t = -1.347741166 + 0.5625 t = -1.347741166 + 0.5625 Combine like terms: -1.347741166 + 0.5625 = -0.785241166 t = -0.785241166 Simplifying t = -0.785241166Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.910241166, -0.785241166}Solution
t = {1.910241166, -0.785241166}
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