9t^2-6t-1=0

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Solution for 9t^2-6t-1=0 equation: 


Simplifying
9t2 + -6t + -1 = 0

Reorder the terms:
-1 + -6t + 9t2 = 0

Solving
-1 + -6t + 9t2 = 0

Solving for variable 't'.

Begin completing the square.  Divide all terms by
9 the coefficient of the squared term: 

Divide each side by '9'.
-0.1111111111 + -0.6666666667t + t2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.6666666667t + 0.1111111111 + t2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.6666666667t + t2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.6666666667t + t2 = 0 + 0.1111111111
-0.6666666667t + t2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.6666666667t + t2 = 0.1111111111

The t term is -0.6666666667t.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667t + 0.1111111112 + t2 = 0.1111111111 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667t + t2 = 0.1111111111 + 0.1111111112

Combine like terms: 0.1111111111 + 0.1111111112 = 0.2222222223
0.1111111112 + -0.6666666667t + t2 = 0.2222222223

Factor a perfect square on the left side:
(t + -0.3333333334)(t + -0.3333333334) = 0.2222222223

Calculate the square root of the right side: 0.471404521

Break this problem into two subproblems by setting 
(t + -0.3333333334) equal to 0.471404521 and -0.471404521.


Subproblem 1
t + -0.3333333334 = 0.471404521

Simplifying
t + -0.3333333334 = 0.471404521

Reorder the terms:
-0.3333333334 + t = 0.471404521

Solving
-0.3333333334 + t = 0.471404521

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + t = 0.471404521 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + t = 0.471404521 + 0.3333333334
t = 0.471404521 + 0.3333333334

Combine like terms: 0.471404521 + 0.3333333334 = 0.8047378544
t = 0.8047378544

Simplifying
t = 0.8047378544


Subproblem 2
t + -0.3333333334 = -0.471404521

Simplifying
t + -0.3333333334 = -0.471404521

Reorder the terms:
-0.3333333334 + t = -0.471404521

Solving
-0.3333333334 + t = -0.471404521

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + t = -0.471404521 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + t = -0.471404521 + 0.3333333334
t = -0.471404521 + 0.3333333334

Combine like terms: -0.471404521 + 0.3333333334 = -0.1380711876
t = -0.1380711876

Simplifying
t = -0.1380711876


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {0.8047378544, -0.1380711876}

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