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Simplifying 75 = -5t2 + -10t + 180 Reorder the terms: 75 = 180 + -10t + -5t2 Solving 75 = 180 + -10t + -5t2 Solving for variable 't'. Combine like terms: 75 + -180 = -105 -105 + 10t + 5t2 = 180 + -10t + -5t2 + -180 + 10t + 5t2 Reorder the terms: -105 + 10t + 5t2 = 180 + -180 + -10t + 10t + -5t2 + 5t2 Combine like terms: 180 + -180 = 0 -105 + 10t + 5t2 = 0 + -10t + 10t + -5t2 + 5t2 -105 + 10t + 5t2 = -10t + 10t + -5t2 + 5t2 Combine like terms: -10t + 10t = 0 -105 + 10t + 5t2 = 0 + -5t2 + 5t2 -105 + 10t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 -105 + 10t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-21 + 2t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-21 + 2t + t2)' equal to zero and attempt to solve: Simplifying -21 + 2t + t2 = 0 Solving -21 + 2t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '21' to each side of the equation. -21 + 2t + 21 + t2 = 0 + 21 Reorder the terms: -21 + 21 + 2t + t2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 2t + t2 = 0 + 21 2t + t2 = 0 + 21 Combine like terms: 0 + 21 = 21 2t + t2 = 21 The t term is 2t. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2t + 1 + t2 = 21 + 1 Reorder the terms: 1 + 2t + t2 = 21 + 1 Combine like terms: 21 + 1 = 22 1 + 2t + t2 = 22 Factor a perfect square on the left side: (t + 1)(t + 1) = 22 Calculate the square root of the right side: 4.69041576 Break this problem into two subproblems by setting (t + 1) equal to 4.69041576 and -4.69041576.Subproblem 1
t + 1 = 4.69041576 Simplifying t + 1 = 4.69041576 Reorder the terms: 1 + t = 4.69041576 Solving 1 + t = 4.69041576 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 4.69041576 + -1 Combine like terms: 1 + -1 = 0 0 + t = 4.69041576 + -1 t = 4.69041576 + -1 Combine like terms: 4.69041576 + -1 = 3.69041576 t = 3.69041576 Simplifying t = 3.69041576Subproblem 2
t + 1 = -4.69041576 Simplifying t + 1 = -4.69041576 Reorder the terms: 1 + t = -4.69041576 Solving 1 + t = -4.69041576 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -4.69041576 + -1 Combine like terms: 1 + -1 = 0 0 + t = -4.69041576 + -1 t = -4.69041576 + -1 Combine like terms: -4.69041576 + -1 = -5.69041576 t = -5.69041576 Simplifying t = -5.69041576Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.69041576, -5.69041576}Solution
t = {3.69041576, -5.69041576}
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