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Simplifying 0 = -5t2 + 10t + 65 Reorder the terms: 0 = 65 + 10t + -5t2 Solving 0 = 65 + 10t + -5t2 Solving for variable 't'. Combine like terms: 0 + -65 = -65 -65 + -10t + 5t2 = 65 + 10t + -5t2 + -65 + -10t + 5t2 Reorder the terms: -65 + -10t + 5t2 = 65 + -65 + 10t + -10t + -5t2 + 5t2 Combine like terms: 65 + -65 = 0 -65 + -10t + 5t2 = 0 + 10t + -10t + -5t2 + 5t2 -65 + -10t + 5t2 = 10t + -10t + -5t2 + 5t2 Combine like terms: 10t + -10t = 0 -65 + -10t + 5t2 = 0 + -5t2 + 5t2 -65 + -10t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 -65 + -10t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-13 + -2t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-13 + -2t + t2)' equal to zero and attempt to solve: Simplifying -13 + -2t + t2 = 0 Solving -13 + -2t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '13' to each side of the equation. -13 + -2t + 13 + t2 = 0 + 13 Reorder the terms: -13 + 13 + -2t + t2 = 0 + 13 Combine like terms: -13 + 13 = 0 0 + -2t + t2 = 0 + 13 -2t + t2 = 0 + 13 Combine like terms: 0 + 13 = 13 -2t + t2 = 13 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 = 13 + 1 Reorder the terms: 1 + -2t + t2 = 13 + 1 Combine like terms: 13 + 1 = 14 1 + -2t + t2 = 14 Factor a perfect square on the left side: (t + -1)(t + -1) = 14 Calculate the square root of the right side: 3.741657387 Break this problem into two subproblems by setting (t + -1) equal to 3.741657387 and -3.741657387.Subproblem 1
t + -1 = 3.741657387 Simplifying t + -1 = 3.741657387 Reorder the terms: -1 + t = 3.741657387 Solving -1 + t = 3.741657387 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 = 3.741657387 + 1 Combine like terms: -1 + 1 = 0 0 + t = 3.741657387 + 1 t = 3.741657387 + 1 Combine like terms: 3.741657387 + 1 = 4.741657387 t = 4.741657387 Simplifying t = 4.741657387Subproblem 2
t + -1 = -3.741657387 Simplifying t + -1 = -3.741657387 Reorder the terms: -1 + t = -3.741657387 Solving -1 + t = -3.741657387 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 = -3.741657387 + 1 Combine like terms: -1 + 1 = 0 0 + t = -3.741657387 + 1 t = -3.741657387 + 1 Combine like terms: -3.741657387 + 1 = -2.741657387 t = -2.741657387 Simplifying t = -2.741657387Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.741657387, -2.741657387}Solution
t = {4.741657387, -2.741657387}
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