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Simplifying 15t2 + 75t + 10 = 0 Reorder the terms: 10 + 75t + 15t2 = 0 Solving 10 + 75t + 15t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '5'. 5(2 + 15t + 3t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(2 + 15t + 3t2)' equal to zero and attempt to solve: Simplifying 2 + 15t + 3t2 = 0 Solving 2 + 15t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + 5t + t2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + 5t + -0.6666666667 + t2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + 5t + t2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + 5t + t2 = 0 + -0.6666666667 5t + t2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 5t + t2 = -0.6666666667 The t term is 5t. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5t + 6.25 + t2 = -0.6666666667 + 6.25 Reorder the terms: 6.25 + 5t + t2 = -0.6666666667 + 6.25 Combine like terms: -0.6666666667 + 6.25 = 5.5833333333 6.25 + 5t + t2 = 5.5833333333 Factor a perfect square on the left side: (t + 2.5)(t + 2.5) = 5.5833333333 Calculate the square root of the right side: 2.362907813 Break this problem into two subproblems by setting (t + 2.5) equal to 2.362907813 and -2.362907813.Subproblem 1
t + 2.5 = 2.362907813 Simplifying t + 2.5 = 2.362907813 Reorder the terms: 2.5 + t = 2.362907813 Solving 2.5 + t = 2.362907813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = 2.362907813 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = 2.362907813 + -2.5 t = 2.362907813 + -2.5 Combine like terms: 2.362907813 + -2.5 = -0.137092187 t = -0.137092187 Simplifying t = -0.137092187Subproblem 2
t + 2.5 = -2.362907813 Simplifying t + 2.5 = -2.362907813 Reorder the terms: 2.5 + t = -2.362907813 Solving 2.5 + t = -2.362907813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + t = -2.362907813 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + t = -2.362907813 + -2.5 t = -2.362907813 + -2.5 Combine like terms: -2.362907813 + -2.5 = -4.862907813 t = -4.862907813 Simplifying t = -4.862907813Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.137092187, -4.862907813}Solution
t = {-0.137092187, -4.862907813}
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