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Simplifying (12t2 + -1t) + -14(12t2 + t) + 13 = 0 Reorder the terms: (-1t + 12t2) + -14(12t2 + t) + 13 = 0 Remove parenthesis around (-1t + 12t2) -1t + 12t2 + -14(12t2 + t) + 13 = 0 Reorder the terms: -1t + 12t2 + -14(t + 12t2) + 13 = 0 -1t + 12t2 + (t * -14 + 12t2 * -14) + 13 = 0 -1t + 12t2 + (-14t + -168t2) + 13 = 0 Reorder the terms: 13 + -1t + -14t + 12t2 + -168t2 = 0 Combine like terms: -1t + -14t = -15t 13 + -15t + 12t2 + -168t2 = 0 Combine like terms: 12t2 + -168t2 = -156t2 13 + -15t + -156t2 = 0 Solving 13 + -15t + -156t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -156 the coefficient of the squared term: Divide each side by '-156'. -0.08333333333 + 0.09615384615t + t2 = 0 Move the constant term to the right: Add '0.08333333333' to each side of the equation. -0.08333333333 + 0.09615384615t + 0.08333333333 + t2 = 0 + 0.08333333333 Reorder the terms: -0.08333333333 + 0.08333333333 + 0.09615384615t + t2 = 0 + 0.08333333333 Combine like terms: -0.08333333333 + 0.08333333333 = 0.00000000000 0.00000000000 + 0.09615384615t + t2 = 0 + 0.08333333333 0.09615384615t + t2 = 0 + 0.08333333333 Combine like terms: 0 + 0.08333333333 = 0.08333333333 0.09615384615t + t2 = 0.08333333333 The t term is 0.09615384615t. Take half its coefficient (0.04807692308). Square it (0.002311390533) and add it to both sides. Add '0.002311390533' to each side of the equation. 0.09615384615t + 0.002311390533 + t2 = 0.08333333333 + 0.002311390533 Reorder the terms: 0.002311390533 + 0.09615384615t + t2 = 0.08333333333 + 0.002311390533 Combine like terms: 0.08333333333 + 0.002311390533 = 0.085644723863 0.002311390533 + 0.09615384615t + t2 = 0.085644723863 Factor a perfect square on the left side: (t + 0.04807692308)(t + 0.04807692308) = 0.085644723863 Calculate the square root of the right side: 0.292651198 Break this problem into two subproblems by setting (t + 0.04807692308) equal to 0.292651198 and -0.292651198.Subproblem 1
t + 0.04807692308 = 0.292651198 Simplifying t + 0.04807692308 = 0.292651198 Reorder the terms: 0.04807692308 + t = 0.292651198 Solving 0.04807692308 + t = 0.292651198 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.04807692308' to each side of the equation. 0.04807692308 + -0.04807692308 + t = 0.292651198 + -0.04807692308 Combine like terms: 0.04807692308 + -0.04807692308 = 0.00000000000 0.00000000000 + t = 0.292651198 + -0.04807692308 t = 0.292651198 + -0.04807692308 Combine like terms: 0.292651198 + -0.04807692308 = 0.24457427492 t = 0.24457427492 Simplifying t = 0.24457427492Subproblem 2
t + 0.04807692308 = -0.292651198 Simplifying t + 0.04807692308 = -0.292651198 Reorder the terms: 0.04807692308 + t = -0.292651198 Solving 0.04807692308 + t = -0.292651198 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.04807692308' to each side of the equation. 0.04807692308 + -0.04807692308 + t = -0.292651198 + -0.04807692308 Combine like terms: 0.04807692308 + -0.04807692308 = 0.00000000000 0.00000000000 + t = -0.292651198 + -0.04807692308 t = -0.292651198 + -0.04807692308 Combine like terms: -0.292651198 + -0.04807692308 = -0.34072812108 t = -0.34072812108 Simplifying t = -0.34072812108Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.24457427492, -0.34072812108}
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