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Simplifying -16t2 + 34t + 80 = t Reorder the terms: 80 + 34t + -16t2 = t Solving 80 + 34t + -16t2 = t Solving for variable 't'. Reorder the terms: 80 + 34t + -1t + -16t2 = t + -1t Combine like terms: 34t + -1t = 33t 80 + 33t + -16t2 = t + -1t Combine like terms: t + -1t = 0 80 + 33t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -5 + -2.0625t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.0625t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -2.0625t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.0625t + t2 = 0 + 5 -2.0625t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.0625t + t2 = 5 The t term is -2.0625t. Take half its coefficient (-1.03125). Square it (1.063476563) and add it to both sides. Add '1.063476563' to each side of the equation. -2.0625t + 1.063476563 + t2 = 5 + 1.063476563 Reorder the terms: 1.063476563 + -2.0625t + t2 = 5 + 1.063476563 Combine like terms: 5 + 1.063476563 = 6.063476563 1.063476563 + -2.0625t + t2 = 6.063476563 Factor a perfect square on the left side: (t + -1.03125)(t + -1.03125) = 6.063476563 Calculate the square root of the right side: 2.462412752 Break this problem into two subproblems by setting (t + -1.03125) equal to 2.462412752 and -2.462412752.Subproblem 1
t + -1.03125 = 2.462412752 Simplifying t + -1.03125 = 2.462412752 Reorder the terms: -1.03125 + t = 2.462412752 Solving -1.03125 + t = 2.462412752 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + t = 2.462412752 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + t = 2.462412752 + 1.03125 t = 2.462412752 + 1.03125 Combine like terms: 2.462412752 + 1.03125 = 3.493662752 t = 3.493662752 Simplifying t = 3.493662752Subproblem 2
t + -1.03125 = -2.462412752 Simplifying t + -1.03125 = -2.462412752 Reorder the terms: -1.03125 + t = -2.462412752 Solving -1.03125 + t = -2.462412752 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + t = -2.462412752 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + t = -2.462412752 + 1.03125 t = -2.462412752 + 1.03125 Combine like terms: -2.462412752 + 1.03125 = -1.431162752 t = -1.431162752 Simplifying t = -1.431162752Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.493662752, -1.431162752}
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