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Simplifying (t2 + -4)(3t2 + -3) + -1(t3 + -3t)(2t) = 0 Reorder the terms: (-4 + t2)(3t2 + -3) + -1(t3 + -3t)(2t) = 0 Reorder the terms: (-4 + t2)(-3 + 3t2) + -1(t3 + -3t)(2t) = 0 Multiply (-4 + t2) * (-3 + 3t2) (-4(-3 + 3t2) + t2(-3 + 3t2)) + -1(t3 + -3t)(2t) = 0 ((-3 * -4 + 3t2 * -4) + t2(-3 + 3t2)) + -1(t3 + -3t)(2t) = 0 ((12 + -12t2) + t2(-3 + 3t2)) + -1(t3 + -3t)(2t) = 0 (12 + -12t2 + (-3 * t2 + 3t2 * t2)) + -1(t3 + -3t)(2t) = 0 (12 + -12t2 + (-3t2 + 3t4)) + -1(t3 + -3t)(2t) = 0 Combine like terms: -12t2 + -3t2 = -15t2 (12 + -15t2 + 3t4) + -1(t3 + -3t)(2t) = 0 Reorder the terms: 12 + -15t2 + 3t4 + -1(-3t + t3)(2t) = 0 Remove parenthesis around (2t) 12 + -15t2 + 3t4 + -1(-3t + t3) * 2t = 0 Reorder the terms for easier multiplication: 12 + -15t2 + 3t4 + -1 * 2t(-3t + t3) = 0 Multiply -1 * 2 12 + -15t2 + 3t4 + -2t(-3t + t3) = 0 12 + -15t2 + 3t4 + (-3t * -2t + t3 * -2t) = 0 12 + -15t2 + 3t4 + (6t2 + -2t4) = 0 Reorder the terms: 12 + -15t2 + 6t2 + 3t4 + -2t4 = 0 Combine like terms: -15t2 + 6t2 = -9t2 12 + -9t2 + 3t4 + -2t4 = 0 Combine like terms: 3t4 + -2t4 = 1t4 12 + -9t2 + 1t4 = 0 Solving 12 + -9t2 + 1t4 = 0 Solving for variable 't'. Begin completing the square. Move the constant term to the right: Add '-12' to each side of the equation. 12 + -9t2 + -12 + t4 = 0 + -12 Reorder the terms: 12 + -12 + -9t2 + t4 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + -9t2 + t4 = 0 + -12 -9t2 + t4 = 0 + -12 Combine like terms: 0 + -12 = -12 -9t2 + t4 = -12 The t term is -9t2. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9t2 + 20.25 + t4 = -12 + 20.25 Reorder the terms: 20.25 + -9t2 + t4 = -12 + 20.25 Combine like terms: -12 + 20.25 = 8.25 20.25 + -9t2 + t4 = 8.25 Factor a perfect square on the left side: (t2 + -4.5)(t2 + -4.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (t2 + -4.5) equal to 2.872281323 and -2.872281323.Subproblem 1
t2 + -4.5 = 2.872281323 Simplifying t2 + -4.5 = 2.872281323 Reorder the terms: -4.5 + t2 = 2.872281323 Solving -4.5 + t2 = 2.872281323 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t2 = 2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t2 = 2.872281323 + 4.5 t2 = 2.872281323 + 4.5 Combine like terms: 2.872281323 + 4.5 = 7.372281323 t2 = 7.372281323 Simplifying t2 = 7.372281323 Take the square root of each side: t = {-2.715194528, 2.715194528}Subproblem 2
t2 + -4.5 = -2.872281323 Simplifying t2 + -4.5 = -2.872281323 Reorder the terms: -4.5 + t2 = -2.872281323 Solving -4.5 + t2 = -2.872281323 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t2 = -2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t2 = -2.872281323 + 4.5 t2 = -2.872281323 + 4.5 Combine like terms: -2.872281323 + 4.5 = 1.627718677 t2 = 1.627718677 Simplifying t2 = 1.627718677 Take the square root of each side: t = {-1.275820786, 1.275820786}Solution
The solution to the problem is based on the solutions from the subproblems. t = {-2.715194528, 2.715194528, -1.275820786, 1.275820786}
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