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Simplifying 9 = 8t(7 + -2t) 9 = (7 * 8t + -2t * 8t) 9 = (56t + -16t2) Solving 9 = 56t + -16t2 Solving for variable 't'. Reorder the terms: 9 + -56t + 16t2 = 56t + -56t + -16t2 + 16t2 Combine like terms: 56t + -56t = 0 9 + -56t + 16t2 = 0 + -16t2 + 16t2 9 + -56t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 9 + -56t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.5625 + -3.5t + t2 = 0 Move the constant term to the right: Add '-0.5625' to each side of the equation. 0.5625 + -3.5t + -0.5625 + t2 = 0 + -0.5625 Reorder the terms: 0.5625 + -0.5625 + -3.5t + t2 = 0 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + -3.5t + t2 = 0 + -0.5625 -3.5t + t2 = 0 + -0.5625 Combine like terms: 0 + -0.5625 = -0.5625 -3.5t + t2 = -0.5625 The t term is -3.5t. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5t + 3.0625 + t2 = -0.5625 + 3.0625 Reorder the terms: 3.0625 + -3.5t + t2 = -0.5625 + 3.0625 Combine like terms: -0.5625 + 3.0625 = 2.5 3.0625 + -3.5t + t2 = 2.5 Factor a perfect square on the left side: (t + -1.75)(t + -1.75) = 2.5 Calculate the square root of the right side: 1.58113883 Break this problem into two subproblems by setting (t + -1.75) equal to 1.58113883 and -1.58113883.Subproblem 1
t + -1.75 = 1.58113883 Simplifying t + -1.75 = 1.58113883 Reorder the terms: -1.75 + t = 1.58113883 Solving -1.75 + t = 1.58113883 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + t = 1.58113883 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + t = 1.58113883 + 1.75 t = 1.58113883 + 1.75 Combine like terms: 1.58113883 + 1.75 = 3.33113883 t = 3.33113883 Simplifying t = 3.33113883Subproblem 2
t + -1.75 = -1.58113883 Simplifying t + -1.75 = -1.58113883 Reorder the terms: -1.75 + t = -1.58113883 Solving -1.75 + t = -1.58113883 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + t = -1.58113883 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + t = -1.58113883 + 1.75 t = -1.58113883 + 1.75 Combine like terms: -1.58113883 + 1.75 = 0.16886117 t = 0.16886117 Simplifying t = 0.16886117Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.33113883, 0.16886117}
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