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Simplifying -16t2 + 144t + 108 = 0 Reorder the terms: 108 + 144t + -16t2 = 0 Solving 108 + 144t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(27 + 36t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(27 + 36t + -4t2)' equal to zero and attempt to solve: Simplifying 27 + 36t + -4t2 = 0 Solving 27 + 36t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -6.75 + -9t + t2 = 0 Move the constant term to the right: Add '6.75' to each side of the equation. -6.75 + -9t + 6.75 + t2 = 0 + 6.75 Reorder the terms: -6.75 + 6.75 + -9t + t2 = 0 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + -9t + t2 = 0 + 6.75 -9t + t2 = 0 + 6.75 Combine like terms: 0 + 6.75 = 6.75 -9t + t2 = 6.75 The t term is -9t. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9t + 20.25 + t2 = 6.75 + 20.25 Reorder the terms: 20.25 + -9t + t2 = 6.75 + 20.25 Combine like terms: 6.75 + 20.25 = 27 20.25 + -9t + t2 = 27 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (t + -4.5) equal to 5.196152423 and -5.196152423.Subproblem 1
t + -4.5 = 5.196152423 Simplifying t + -4.5 = 5.196152423 Reorder the terms: -4.5 + t = 5.196152423 Solving -4.5 + t = 5.196152423 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = 5.196152423 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 5.196152423 + 4.5 t = 5.196152423 + 4.5 Combine like terms: 5.196152423 + 4.5 = 9.696152423 t = 9.696152423 Simplifying t = 9.696152423Subproblem 2
t + -4.5 = -5.196152423 Simplifying t + -4.5 = -5.196152423 Reorder the terms: -4.5 + t = -5.196152423 Solving -4.5 + t = -5.196152423 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = -5.196152423 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -5.196152423 + 4.5 t = -5.196152423 + 4.5 Combine like terms: -5.196152423 + 4.5 = -0.696152423 t = -0.696152423 Simplifying t = -0.696152423Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.696152423, -0.696152423}Solution
t = {9.696152423, -0.696152423}
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