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Simplifying -16t2 + 128t + 180 = 0 Reorder the terms: 180 + 128t + -16t2 = 0 Solving 180 + 128t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(45 + 32t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(45 + 32t + -4t2)' equal to zero and attempt to solve: Simplifying 45 + 32t + -4t2 = 0 Solving 45 + 32t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -11.25 + -8t + t2 = 0 Move the constant term to the right: Add '11.25' to each side of the equation. -11.25 + -8t + 11.25 + t2 = 0 + 11.25 Reorder the terms: -11.25 + 11.25 + -8t + t2 = 0 + 11.25 Combine like terms: -11.25 + 11.25 = 0.00 0.00 + -8t + t2 = 0 + 11.25 -8t + t2 = 0 + 11.25 Combine like terms: 0 + 11.25 = 11.25 -8t + t2 = 11.25 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = 11.25 + 16 Reorder the terms: 16 + -8t + t2 = 11.25 + 16 Combine like terms: 11.25 + 16 = 27.25 16 + -8t + t2 = 27.25 Factor a perfect square on the left side: (t + -4)(t + -4) = 27.25 Calculate the square root of the right side: 5.220153254 Break this problem into two subproblems by setting (t + -4) equal to 5.220153254 and -5.220153254.Subproblem 1
t + -4 = 5.220153254 Simplifying t + -4 = 5.220153254 Reorder the terms: -4 + t = 5.220153254 Solving -4 + t = 5.220153254 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 5.220153254 + 4 Combine like terms: -4 + 4 = 0 0 + t = 5.220153254 + 4 t = 5.220153254 + 4 Combine like terms: 5.220153254 + 4 = 9.220153254 t = 9.220153254 Simplifying t = 9.220153254Subproblem 2
t + -4 = -5.220153254 Simplifying t + -4 = -5.220153254 Reorder the terms: -4 + t = -5.220153254 Solving -4 + t = -5.220153254 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -5.220153254 + 4 Combine like terms: -4 + 4 = 0 0 + t = -5.220153254 + 4 t = -5.220153254 + 4 Combine like terms: -5.220153254 + 4 = -1.220153254 t = -1.220153254 Simplifying t = -1.220153254Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.220153254, -1.220153254}Solution
t = {9.220153254, -1.220153254}
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