7a^2-2a-6=0

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Solution for 7a^2-2a-6=0 equation: 


Simplifying
7a2 + -2a + -6 = 0

Reorder the terms:
-6 + -2a + 7a2 = 0

Solving
-6 + -2a + 7a2 = 0

Solving for variable 'a'.

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

Divide each side by '7'.
-0.8571428571 + -0.2857142857a + a2 = 0

Move the constant term to the right:

Add '0.8571428571' to each side of the equation.
-0.8571428571 + -0.2857142857a + 0.8571428571 + a2 = 0 + 0.8571428571

Reorder the terms:
-0.8571428571 + 0.8571428571 + -0.2857142857a + a2 = 0 + 0.8571428571

Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000
0.0000000000 + -0.2857142857a + a2 = 0 + 0.8571428571
-0.2857142857a + a2 = 0 + 0.8571428571

Combine like terms: 0 + 0.8571428571 = 0.8571428571
-0.2857142857a + a2 = 0.8571428571

The a term is -0.2857142857a.  Take half its coefficient (-0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
-0.2857142857a + 0.02040816328 + a2 = 0.8571428571 + 0.02040816328

Reorder the terms:
0.02040816328 + -0.2857142857a + a2 = 0.8571428571 + 0.02040816328

Combine like terms: 0.8571428571 + 0.02040816328 = 0.87755102038
0.02040816328 + -0.2857142857a + a2 = 0.87755102038

Factor a perfect square on the left side:
(a + -0.1428571429)(a + -0.1428571429) = 0.87755102038

Calculate the square root of the right side: 0.936776932

Break this problem into two subproblems by setting 
(a + -0.1428571429) equal to 0.936776932 and -0.936776932.


Subproblem 1
a + -0.1428571429 = 0.936776932

Simplifying
a + -0.1428571429 = 0.936776932

Reorder the terms:
-0.1428571429 + a = 0.936776932

Solving
-0.1428571429 + a = 0.936776932

Solving for variable 'a'.

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

Add '0.1428571429' to each side of the equation.
-0.1428571429 + 0.1428571429 + a = 0.936776932 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + a = 0.936776932 + 0.1428571429
a = 0.936776932 + 0.1428571429

Combine like terms: 0.936776932 + 0.1428571429 = 1.0796340749
a = 1.0796340749

Simplifying
a = 1.0796340749


Subproblem 2
a + -0.1428571429 = -0.936776932

Simplifying
a + -0.1428571429 = -0.936776932

Reorder the terms:
-0.1428571429 + a = -0.936776932

Solving
-0.1428571429 + a = -0.936776932

Solving for variable 'a'.

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

Add '0.1428571429' to each side of the equation.
-0.1428571429 + 0.1428571429 + a = -0.936776932 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + a = -0.936776932 + 0.1428571429
a = -0.936776932 + 0.1428571429

Combine like terms: -0.936776932 + 0.1428571429 = -0.7939197891
a = -0.7939197891

Simplifying
a = -0.7939197891


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {1.0796340749, -0.7939197891}

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