-16t^2+34t+80=t

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Solution for -16t^2+34t+80=t equation:


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 = 3.493662752

Subproblem 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 = -1.431162752

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.493662752, -1.431162752}

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