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Simplifying 9t2 + 14 + -17t + 6t + -8t2 = 0 Reorder the terms: 14 + -17t + 6t + 9t2 + -8t2 = 0 Combine like terms: -17t + 6t = -11t 14 + -11t + 9t2 + -8t2 = 0 Combine like terms: 9t2 + -8t2 = 1t2 14 + -11t + 1t2 = 0 Solving 14 + -11t + 1t2 = 0 Solving for variable 't'. Begin completing the square. Move the constant term to the right: Add '-14' to each side of the equation. 14 + -11t + -14 + t2 = 0 + -14 Reorder the terms: 14 + -14 + -11t + t2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + -11t + t2 = 0 + -14 -11t + t2 = 0 + -14 Combine like terms: 0 + -14 = -14 -11t + t2 = -14 The t term is -11t. Take half its coefficient (-5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. -11t + 30.25 + t2 = -14 + 30.25 Reorder the terms: 30.25 + -11t + t2 = -14 + 30.25 Combine like terms: -14 + 30.25 = 16.25 30.25 + -11t + t2 = 16.25 Factor a perfect square on the left side: (t + -5.5)(t + -5.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (t + -5.5) equal to 4.031128874 and -4.031128874.Subproblem 1
t + -5.5 = 4.031128874 Simplifying t + -5.5 = 4.031128874 Reorder the terms: -5.5 + t = 4.031128874 Solving -5.5 + t = 4.031128874 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + t = 4.031128874 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + t = 4.031128874 + 5.5 t = 4.031128874 + 5.5 Combine like terms: 4.031128874 + 5.5 = 9.531128874 t = 9.531128874 Simplifying t = 9.531128874Subproblem 2
t + -5.5 = -4.031128874 Simplifying t + -5.5 = -4.031128874 Reorder the terms: -5.5 + t = -4.031128874 Solving -5.5 + t = -4.031128874 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + t = -4.031128874 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + t = -4.031128874 + 5.5 t = -4.031128874 + 5.5 Combine like terms: -4.031128874 + 5.5 = 1.468871126 t = 1.468871126 Simplifying t = 1.468871126Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.531128874, 1.468871126}
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