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Simplifying 5(t + -2) * 7t = 6(2t + 3) + -9 Reorder the terms: 5(-2 + t) * 7t = 6(2t + 3) + -9 Reorder the terms for easier multiplication: 5 * 7t(-2 + t) = 6(2t + 3) + -9 Multiply 5 * 7 35t(-2 + t) = 6(2t + 3) + -9 (-2 * 35t + t * 35t) = 6(2t + 3) + -9 (-70t + 35t2) = 6(2t + 3) + -9 Reorder the terms: -70t + 35t2 = 6(3 + 2t) + -9 -70t + 35t2 = (3 * 6 + 2t * 6) + -9 -70t + 35t2 = (18 + 12t) + -9 Reorder the terms: -70t + 35t2 = 18 + -9 + 12t Combine like terms: 18 + -9 = 9 -70t + 35t2 = 9 + 12t Solving -70t + 35t2 = 9 + 12t Solving for variable 't'. Reorder the terms: -9 + -70t + -12t + 35t2 = 9 + 12t + -9 + -12t Combine like terms: -70t + -12t = -82t -9 + -82t + 35t2 = 9 + 12t + -9 + -12t Reorder the terms: -9 + -82t + 35t2 = 9 + -9 + 12t + -12t Combine like terms: 9 + -9 = 0 -9 + -82t + 35t2 = 0 + 12t + -12t -9 + -82t + 35t2 = 12t + -12t Combine like terms: 12t + -12t = 0 -9 + -82t + 35t2 = 0 Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. -0.2571428571 + -2.342857143t + t2 = 0 Move the constant term to the right: Add '0.2571428571' to each side of the equation. -0.2571428571 + -2.342857143t + 0.2571428571 + t2 = 0 + 0.2571428571 Reorder the terms: -0.2571428571 + 0.2571428571 + -2.342857143t + t2 = 0 + 0.2571428571 Combine like terms: -0.2571428571 + 0.2571428571 = 0.0000000000 0.0000000000 + -2.342857143t + t2 = 0 + 0.2571428571 -2.342857143t + t2 = 0 + 0.2571428571 Combine like terms: 0 + 0.2571428571 = 0.2571428571 -2.342857143t + t2 = 0.2571428571 The t term is -2.342857143t. Take half its coefficient (-1.171428572). Square it (1.372244899) and add it to both sides. Add '1.372244899' to each side of the equation. -2.342857143t + 1.372244899 + t2 = 0.2571428571 + 1.372244899 Reorder the terms: 1.372244899 + -2.342857143t + t2 = 0.2571428571 + 1.372244899 Combine like terms: 0.2571428571 + 1.372244899 = 1.6293877561 1.372244899 + -2.342857143t + t2 = 1.6293877561 Factor a perfect square on the left side: (t + -1.171428572)(t + -1.171428572) = 1.6293877561 Calculate the square root of the right side: 1.276474738 Break this problem into two subproblems by setting (t + -1.171428572) equal to 1.276474738 and -1.276474738.Subproblem 1
t + -1.171428572 = 1.276474738 Simplifying t + -1.171428572 = 1.276474738 Reorder the terms: -1.171428572 + t = 1.276474738 Solving -1.171428572 + t = 1.276474738 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.171428572' to each side of the equation. -1.171428572 + 1.171428572 + t = 1.276474738 + 1.171428572 Combine like terms: -1.171428572 + 1.171428572 = 0.000000000 0.000000000 + t = 1.276474738 + 1.171428572 t = 1.276474738 + 1.171428572 Combine like terms: 1.276474738 + 1.171428572 = 2.44790331 t = 2.44790331 Simplifying t = 2.44790331Subproblem 2
t + -1.171428572 = -1.276474738 Simplifying t + -1.171428572 = -1.276474738 Reorder the terms: -1.171428572 + t = -1.276474738 Solving -1.171428572 + t = -1.276474738 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.171428572' to each side of the equation. -1.171428572 + 1.171428572 + t = -1.276474738 + 1.171428572 Combine like terms: -1.171428572 + 1.171428572 = 0.000000000 0.000000000 + t = -1.276474738 + 1.171428572 t = -1.276474738 + 1.171428572 Combine like terms: -1.276474738 + 1.171428572 = -0.105046166 t = -0.105046166 Simplifying t = -0.105046166Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.44790331, -0.105046166}
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