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Simplifying 4(t2 + -1) = 3t2 + 6t + 5 Reorder the terms: 4(-1 + t2) = 3t2 + 6t + 5 (-1 * 4 + t2 * 4) = 3t2 + 6t + 5 (-4 + 4t2) = 3t2 + 6t + 5 Reorder the terms: -4 + 4t2 = 5 + 6t + 3t2 Solving -4 + 4t2 = 5 + 6t + 3t2 Solving for variable 't'. Reorder the terms: -4 + -5 + -6t + 4t2 + -3t2 = 5 + 6t + 3t2 + -5 + -6t + -3t2 Combine like terms: -4 + -5 = -9 -9 + -6t + 4t2 + -3t2 = 5 + 6t + 3t2 + -5 + -6t + -3t2 Combine like terms: 4t2 + -3t2 = 1t2 -9 + -6t + 1t2 = 5 + 6t + 3t2 + -5 + -6t + -3t2 Reorder the terms: -9 + -6t + 1t2 = 5 + -5 + 6t + -6t + 3t2 + -3t2 Combine like terms: 5 + -5 = 0 -9 + -6t + 1t2 = 0 + 6t + -6t + 3t2 + -3t2 -9 + -6t + 1t2 = 6t + -6t + 3t2 + -3t2 Combine like terms: 6t + -6t = 0 -9 + -6t + 1t2 = 0 + 3t2 + -3t2 -9 + -6t + 1t2 = 3t2 + -3t2 Combine like terms: 3t2 + -3t2 = 0 -9 + -6t + 1t2 = 0 Begin completing the square. Move the constant term to the right: Add '9' to each side of the equation. -9 + -6t + 9 + t2 = 0 + 9 Reorder the terms: -9 + 9 + -6t + t2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -6t + t2 = 0 + 9 -6t + t2 = 0 + 9 Combine like terms: 0 + 9 = 9 -6t + t2 = 9 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 = 9 + 9 Reorder the terms: 9 + -6t + t2 = 9 + 9 Combine like terms: 9 + 9 = 18 9 + -6t + t2 = 18 Factor a perfect square on the left side: (t + -3)(t + -3) = 18 Calculate the square root of the right side: 4.242640687 Break this problem into two subproblems by setting (t + -3) equal to 4.242640687 and -4.242640687.Subproblem 1
t + -3 = 4.242640687 Simplifying t + -3 = 4.242640687 Reorder the terms: -3 + t = 4.242640687 Solving -3 + t = 4.242640687 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 = 4.242640687 + 3 Combine like terms: -3 + 3 = 0 0 + t = 4.242640687 + 3 t = 4.242640687 + 3 Combine like terms: 4.242640687 + 3 = 7.242640687 t = 7.242640687 Simplifying t = 7.242640687Subproblem 2
t + -3 = -4.242640687 Simplifying t + -3 = -4.242640687 Reorder the terms: -3 + t = -4.242640687 Solving -3 + t = -4.242640687 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 = -4.242640687 + 3 Combine like terms: -3 + 3 = 0 0 + t = -4.242640687 + 3 t = -4.242640687 + 3 Combine like terms: -4.242640687 + 3 = -1.242640687 t = -1.242640687 Simplifying t = -1.242640687Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.242640687, -1.242640687}
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