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Simplifying 175 = (2t + 7)(2t + 5) Reorder the terms: 175 = (7 + 2t)(2t + 5) Reorder the terms: 175 = (7 + 2t)(5 + 2t) Multiply (7 + 2t) * (5 + 2t) 175 = (7(5 + 2t) + 2t * (5 + 2t)) 175 = ((5 * 7 + 2t * 7) + 2t * (5 + 2t)) 175 = ((35 + 14t) + 2t * (5 + 2t)) 175 = (35 + 14t + (5 * 2t + 2t * 2t)) 175 = (35 + 14t + (10t + 4t2)) Combine like terms: 14t + 10t = 24t 175 = (35 + 24t + 4t2) Solving 175 = 35 + 24t + 4t2 Solving for variable 't'. Combine like terms: 175 + -35 = 140 140 + -24t + -4t2 = 35 + 24t + 4t2 + -35 + -24t + -4t2 Reorder the terms: 140 + -24t + -4t2 = 35 + -35 + 24t + -24t + 4t2 + -4t2 Combine like terms: 35 + -35 = 0 140 + -24t + -4t2 = 0 + 24t + -24t + 4t2 + -4t2 140 + -24t + -4t2 = 24t + -24t + 4t2 + -4t2 Combine like terms: 24t + -24t = 0 140 + -24t + -4t2 = 0 + 4t2 + -4t2 140 + -24t + -4t2 = 4t2 + -4t2 Combine like terms: 4t2 + -4t2 = 0 140 + -24t + -4t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(35 + -6t + -1t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(35 + -6t + -1t2)' equal to zero and attempt to solve: Simplifying 35 + -6t + -1t2 = 0 Solving 35 + -6t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -35 + 6t + t2 = 0 Move the constant term to the right: Add '35' to each side of the equation. -35 + 6t + 35 + t2 = 0 + 35 Reorder the terms: -35 + 35 + 6t + t2 = 0 + 35 Combine like terms: -35 + 35 = 0 0 + 6t + t2 = 0 + 35 6t + t2 = 0 + 35 Combine like terms: 0 + 35 = 35 6t + t2 = 35 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 = 35 + 9 Reorder the terms: 9 + 6t + t2 = 35 + 9 Combine like terms: 35 + 9 = 44 9 + 6t + t2 = 44 Factor a perfect square on the left side: (t + 3)(t + 3) = 44 Calculate the square root of the right side: 6.633249581 Break this problem into two subproblems by setting (t + 3) equal to 6.633249581 and -6.633249581.Subproblem 1
t + 3 = 6.633249581 Simplifying t + 3 = 6.633249581 Reorder the terms: 3 + t = 6.633249581 Solving 3 + t = 6.633249581 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 = 6.633249581 + -3 Combine like terms: 3 + -3 = 0 0 + t = 6.633249581 + -3 t = 6.633249581 + -3 Combine like terms: 6.633249581 + -3 = 3.633249581 t = 3.633249581 Simplifying t = 3.633249581Subproblem 2
t + 3 = -6.633249581 Simplifying t + 3 = -6.633249581 Reorder the terms: 3 + t = -6.633249581 Solving 3 + t = -6.633249581 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 = -6.633249581 + -3 Combine like terms: 3 + -3 = 0 0 + t = -6.633249581 + -3 t = -6.633249581 + -3 Combine like terms: -6.633249581 + -3 = -9.633249581 t = -9.633249581 Simplifying t = -9.633249581Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.633249581, -9.633249581}Solution
t = {3.633249581, -9.633249581}
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