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Simplifying 5(t2 + -1) = 4t2 + 8t + 7 Reorder the terms: 5(-1 + t2) = 4t2 + 8t + 7 (-1 * 5 + t2 * 5) = 4t2 + 8t + 7 (-5 + 5t2) = 4t2 + 8t + 7 Reorder the terms: -5 + 5t2 = 7 + 8t + 4t2 Solving -5 + 5t2 = 7 + 8t + 4t2 Solving for variable 't'. Reorder the terms: -5 + -7 + -8t + 5t2 + -4t2 = 7 + 8t + 4t2 + -7 + -8t + -4t2 Combine like terms: -5 + -7 = -12 -12 + -8t + 5t2 + -4t2 = 7 + 8t + 4t2 + -7 + -8t + -4t2 Combine like terms: 5t2 + -4t2 = 1t2 -12 + -8t + 1t2 = 7 + 8t + 4t2 + -7 + -8t + -4t2 Reorder the terms: -12 + -8t + 1t2 = 7 + -7 + 8t + -8t + 4t2 + -4t2 Combine like terms: 7 + -7 = 0 -12 + -8t + 1t2 = 0 + 8t + -8t + 4t2 + -4t2 -12 + -8t + 1t2 = 8t + -8t + 4t2 + -4t2 Combine like terms: 8t + -8t = 0 -12 + -8t + 1t2 = 0 + 4t2 + -4t2 -12 + -8t + 1t2 = 4t2 + -4t2 Combine like terms: 4t2 + -4t2 = 0 -12 + -8t + 1t2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + -8t + 12 + t2 = 0 + 12 Reorder the terms: -12 + 12 + -8t + t2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -8t + t2 = 0 + 12 -8t + t2 = 0 + 12 Combine like terms: 0 + 12 = 12 -8t + t2 = 12 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = 12 + 16 Reorder the terms: 16 + -8t + t2 = 12 + 16 Combine like terms: 12 + 16 = 28 16 + -8t + t2 = 28 Factor a perfect square on the left side: (t + -4)(t + -4) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (t + -4) equal to 5.291502622 and -5.291502622.Subproblem 1
t + -4 = 5.291502622 Simplifying t + -4 = 5.291502622 Reorder the terms: -4 + t = 5.291502622 Solving -4 + t = 5.291502622 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 5.291502622 + 4 Combine like terms: -4 + 4 = 0 0 + t = 5.291502622 + 4 t = 5.291502622 + 4 Combine like terms: 5.291502622 + 4 = 9.291502622 t = 9.291502622 Simplifying t = 9.291502622Subproblem 2
t + -4 = -5.291502622 Simplifying t + -4 = -5.291502622 Reorder the terms: -4 + t = -5.291502622 Solving -4 + t = -5.291502622 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -5.291502622 + 4 Combine like terms: -4 + 4 = 0 0 + t = -5.291502622 + 4 t = -5.291502622 + 4 Combine like terms: -5.291502622 + 4 = -1.291502622 t = -1.291502622 Simplifying t = -1.291502622Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.291502622, -1.291502622}
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