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Simplifying 128t + -16t2 + -40 = 0 Reorder the terms: -40 + 128t + -16t2 = 0 Solving -40 + 128t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '8'. 8(-5 + 16t + -2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-5 + 16t + -2t2)' equal to zero and attempt to solve: Simplifying -5 + 16t + -2t2 = 0 Solving -5 + 16t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 2.5 + -8t + t2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -8t + -2.5 + t2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -8t + t2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -8t + t2 = 0 + -2.5 -8t + t2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -8t + t2 = -2.5 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 = -2.5 + 16 Reorder the terms: 16 + -8t + t2 = -2.5 + 16 Combine like terms: -2.5 + 16 = 13.5 16 + -8t + t2 = 13.5 Factor a perfect square on the left side: (t + -4)(t + -4) = 13.5 Calculate the square root of the right side: 3.674234614 Break this problem into two subproblems by setting (t + -4) equal to 3.674234614 and -3.674234614.Subproblem 1
t + -4 = 3.674234614 Simplifying t + -4 = 3.674234614 Reorder the terms: -4 + t = 3.674234614 Solving -4 + t = 3.674234614 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 = 3.674234614 + 4 Combine like terms: -4 + 4 = 0 0 + t = 3.674234614 + 4 t = 3.674234614 + 4 Combine like terms: 3.674234614 + 4 = 7.674234614 t = 7.674234614 Simplifying t = 7.674234614Subproblem 2
t + -4 = -3.674234614 Simplifying t + -4 = -3.674234614 Reorder the terms: -4 + t = -3.674234614 Solving -4 + t = -3.674234614 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 = -3.674234614 + 4 Combine like terms: -4 + 4 = 0 0 + t = -3.674234614 + 4 t = -3.674234614 + 4 Combine like terms: -3.674234614 + 4 = 0.325765386 t = 0.325765386 Simplifying t = 0.325765386Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.674234614, 0.325765386}Solution
t = {7.674234614, 0.325765386}
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