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Simplifying 15t2 + 90t + 130 = 0 Reorder the terms: 130 + 90t + 15t2 = 0 Solving 130 + 90t + 15t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '5'. 5(26 + 18t + 3t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(26 + 18t + 3t2)' equal to zero and attempt to solve: Simplifying 26 + 18t + 3t2 = 0 Solving 26 + 18t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 8.666666667 + 6t + t2 = 0 Move the constant term to the right: Add '-8.666666667' to each side of the equation. 8.666666667 + 6t + -8.666666667 + t2 = 0 + -8.666666667 Reorder the terms: 8.666666667 + -8.666666667 + 6t + t2 = 0 + -8.666666667 Combine like terms: 8.666666667 + -8.666666667 = 0.000000000 0.000000000 + 6t + t2 = 0 + -8.666666667 6t + t2 = 0 + -8.666666667 Combine like terms: 0 + -8.666666667 = -8.666666667 6t + t2 = -8.666666667 The t term is 6t. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6t + 9 + t2 = -8.666666667 + 9 Reorder the terms: 9 + 6t + t2 = -8.666666667 + 9 Combine like terms: -8.666666667 + 9 = 0.333333333 9 + 6t + t2 = 0.333333333 Factor a perfect square on the left side: (t + 3)(t + 3) = 0.333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (t + 3) equal to 0.577350269 and -0.577350269.Subproblem 1
t + 3 = 0.577350269 Simplifying t + 3 = 0.577350269 Reorder the terms: 3 + t = 0.577350269 Solving 3 + t = 0.577350269 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = 0.577350269 + -3 Combine like terms: 3 + -3 = 0 0 + t = 0.577350269 + -3 t = 0.577350269 + -3 Combine like terms: 0.577350269 + -3 = -2.422649731 t = -2.422649731 Simplifying t = -2.422649731Subproblem 2
t + 3 = -0.577350269 Simplifying t + 3 = -0.577350269 Reorder the terms: 3 + t = -0.577350269 Solving 3 + t = -0.577350269 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = -0.577350269 + -3 Combine like terms: 3 + -3 = 0 0 + t = -0.577350269 + -3 t = -0.577350269 + -3 Combine like terms: -0.577350269 + -3 = -3.577350269 t = -3.577350269 Simplifying t = -3.577350269Solution
The solution to the problem is based on the solutions from the subproblems. t = {-2.422649731, -3.577350269}Solution
t = {-2.422649731, -3.577350269}
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