6t^2-5t-7=0

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Solution for 6t^2-5t-7=0 equation: 


Simplifying
6t2 + -5t + -7 = 0

Reorder the terms:
-7 + -5t + 6t2 = 0

Solving
-7 + -5t + 6t2 = 0

Solving for variable 't'.

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

Divide each side by '6'.
-1.166666667 + -0.8333333333t + t2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + -0.8333333333t + 1.166666667 + t2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + -0.8333333333t + t2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + -0.8333333333t + t2 = 0 + 1.166666667
-0.8333333333t + t2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
-0.8333333333t + t2 = 1.166666667

The t term is -0.8333333333t.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333t + 0.1736111111 + t2 = 1.166666667 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333t + t2 = 1.166666667 + 0.1736111111

Combine like terms: 1.166666667 + 0.1736111111 = 1.3402777781
0.1736111111 + -0.8333333333t + t2 = 1.3402777781

Factor a perfect square on the left side:
(t + -0.4166666667)(t + -0.4166666667) = 1.3402777781

Calculate the square root of the right side: 1.157703666

Break this problem into two subproblems by setting 
(t + -0.4166666667) equal to 1.157703666 and -1.157703666.


Subproblem 1
t + -0.4166666667 = 1.157703666

Simplifying
t + -0.4166666667 = 1.157703666

Reorder the terms:
-0.4166666667 + t = 1.157703666

Solving
-0.4166666667 + t = 1.157703666

Solving for variable 't'.

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

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.4166666667 + t = 1.157703666 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + t = 1.157703666 + 0.4166666667
t = 1.157703666 + 0.4166666667

Combine like terms: 1.157703666 + 0.4166666667 = 1.5743703327
t = 1.5743703327

Simplifying
t = 1.5743703327


Subproblem 2
t + -0.4166666667 = -1.157703666

Simplifying
t + -0.4166666667 = -1.157703666

Reorder the terms:
-0.4166666667 + t = -1.157703666

Solving
-0.4166666667 + t = -1.157703666

Solving for variable 't'.

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

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.4166666667 + t = -1.157703666 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + t = -1.157703666 + 0.4166666667
t = -1.157703666 + 0.4166666667

Combine like terms: -1.157703666 + 0.4166666667 = -0.7410369993
t = -0.7410369993

Simplifying
t = -0.7410369993


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.5743703327, -0.7410369993}

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