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Simplifying 3(3t + -4) + -1(t2 + 5) + -4t(t + -3) = 0 Reorder the terms: 3(-4 + 3t) + -1(t2 + 5) + -4t(t + -3) = 0 (-4 * 3 + 3t * 3) + -1(t2 + 5) + -4t(t + -3) = 0 (-12 + 9t) + -1(t2 + 5) + -4t(t + -3) = 0 Reorder the terms: -12 + 9t + -1(5 + t2) + -4t(t + -3) = 0 -12 + 9t + (5 * -1 + t2 * -1) + -4t(t + -3) = 0 -12 + 9t + (-5 + -1t2) + -4t(t + -3) = 0 Reorder the terms: -12 + 9t + -5 + -1t2 + -4t(-3 + t) = 0 -12 + 9t + -5 + -1t2 + (-3 * -4t + t * -4t) = 0 -12 + 9t + -5 + -1t2 + (12t + -4t2) = 0 Reorder the terms: -12 + -5 + 9t + 12t + -1t2 + -4t2 = 0 Combine like terms: -12 + -5 = -17 -17 + 9t + 12t + -1t2 + -4t2 = 0 Combine like terms: 9t + 12t = 21t -17 + 21t + -1t2 + -4t2 = 0 Combine like terms: -1t2 + -4t2 = -5t2 -17 + 21t + -5t2 = 0 Solving -17 + 21t + -5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 3.4 + -4.2t + t2 = 0 Move the constant term to the right: Add '-3.4' to each side of the equation. 3.4 + -4.2t + -3.4 + t2 = 0 + -3.4 Reorder the terms: 3.4 + -3.4 + -4.2t + t2 = 0 + -3.4 Combine like terms: 3.4 + -3.4 = 0.0 0.0 + -4.2t + t2 = 0 + -3.4 -4.2t + t2 = 0 + -3.4 Combine like terms: 0 + -3.4 = -3.4 -4.2t + t2 = -3.4 The t term is -4.2t. Take half its coefficient (-2.1). Square it (4.41) and add it to both sides. Add '4.41' to each side of the equation. -4.2t + 4.41 + t2 = -3.4 + 4.41 Reorder the terms: 4.41 + -4.2t + t2 = -3.4 + 4.41 Combine like terms: -3.4 + 4.41 = 1.01 4.41 + -4.2t + t2 = 1.01 Factor a perfect square on the left side: (t + -2.1)(t + -2.1) = 1.01 Calculate the square root of the right side: 1.004987562 Break this problem into two subproblems by setting (t + -2.1) equal to 1.004987562 and -1.004987562.Subproblem 1
t + -2.1 = 1.004987562 Simplifying t + -2.1 = 1.004987562 Reorder the terms: -2.1 + t = 1.004987562 Solving -2.1 + t = 1.004987562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.1' to each side of the equation. -2.1 + 2.1 + t = 1.004987562 + 2.1 Combine like terms: -2.1 + 2.1 = 0.0 0.0 + t = 1.004987562 + 2.1 t = 1.004987562 + 2.1 Combine like terms: 1.004987562 + 2.1 = 3.104987562 t = 3.104987562 Simplifying t = 3.104987562Subproblem 2
t + -2.1 = -1.004987562 Simplifying t + -2.1 = -1.004987562 Reorder the terms: -2.1 + t = -1.004987562 Solving -2.1 + t = -1.004987562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.1' to each side of the equation. -2.1 + 2.1 + t = -1.004987562 + 2.1 Combine like terms: -2.1 + 2.1 = 0.0 0.0 + t = -1.004987562 + 2.1 t = -1.004987562 + 2.1 Combine like terms: -1.004987562 + 2.1 = 1.095012438 t = 1.095012438 Simplifying t = 1.095012438Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.104987562, 1.095012438}
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