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Simplifying (3t2 + -2t + 7) + (-4t2 + 5t + 1) = 0 Reorder the terms: (7 + -2t + 3t2) + (-4t2 + 5t + 1) = 0 Remove parenthesis around (7 + -2t + 3t2) 7 + -2t + 3t2 + (-4t2 + 5t + 1) = 0 Reorder the terms: 7 + -2t + 3t2 + (1 + 5t + -4t2) = 0 Remove parenthesis around (1 + 5t + -4t2) 7 + -2t + 3t2 + 1 + 5t + -4t2 = 0 Reorder the terms: 7 + 1 + -2t + 5t + 3t2 + -4t2 = 0 Combine like terms: 7 + 1 = 8 8 + -2t + 5t + 3t2 + -4t2 = 0 Combine like terms: -2t + 5t = 3t 8 + 3t + 3t2 + -4t2 = 0 Combine like terms: 3t2 + -4t2 = -1t2 8 + 3t + -1t2 = 0 Solving 8 + 3t + -1t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -8 + -3t + t2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -3t + 8 + t2 = 0 + 8 Reorder the terms: -8 + 8 + -3t + t2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -3t + t2 = 0 + 8 -3t + t2 = 0 + 8 Combine like terms: 0 + 8 = 8 -3t + t2 = 8 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = 8 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 8 + 2.25 Combine like terms: 8 + 2.25 = 10.25 2.25 + -3t + t2 = 10.25 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 10.25 Calculate the square root of the right side: 3.201562119 Break this problem into two subproblems by setting (t + -1.5) equal to 3.201562119 and -3.201562119.Subproblem 1
t + -1.5 = 3.201562119 Simplifying t + -1.5 = 3.201562119 Reorder the terms: -1.5 + t = 3.201562119 Solving -1.5 + t = 3.201562119 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 3.201562119 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 3.201562119 + 1.5 t = 3.201562119 + 1.5 Combine like terms: 3.201562119 + 1.5 = 4.701562119 t = 4.701562119 Simplifying t = 4.701562119Subproblem 2
t + -1.5 = -3.201562119 Simplifying t + -1.5 = -3.201562119 Reorder the terms: -1.5 + t = -3.201562119 Solving -1.5 + t = -3.201562119 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -3.201562119 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -3.201562119 + 1.5 t = -3.201562119 + 1.5 Combine like terms: -3.201562119 + 1.5 = -1.701562119 t = -1.701562119 Simplifying t = -1.701562119Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.701562119, -1.701562119}
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