x+4a+-3a^2=11a-3+x

Solution for x+4a+-3a^2=11a-3+x equation:

Simplifying
x + 4a + -3a2 = 11a + -3 + x

Reorder the terms:
4a + -3a2 + x = 11a + -3 + x

Reorder the terms:
4a + -3a2 + x = -3 + 11a + x

Add '-1x' to each side of the equation.
4a + -3a2 + x + -1x = -3 + 11a + x + -1x

Combine like terms: x + -1x = 0
4a + -3a2 + 0 = -3 + 11a + x + -1x
4a + -3a2 = -3 + 11a + x + -1x

Combine like terms: x + -1x = 0
4a + -3a2 = -3 + 11a + 0
4a + -3a2 = -3 + 11a

Solving
4a + -3a2 = -3 + 11a

Solving for variable 'a'.

Reorder the terms:
3 + 4a + -11a + -3a2 = -3 + 11a + 3 + -11a

Combine like terms: 4a + -11a = -7a
3 + -7a + -3a2 = -3 + 11a + 3 + -11a

Reorder the terms:
3 + -7a + -3a2 = -3 + 3 + 11a + -11a

Combine like terms: -3 + 3 = 0
3 + -7a + -3a2 = 0 + 11a + -11a
3 + -7a + -3a2 = 11a + -11a

Combine like terms: 11a + -11a = 0
3 + -7a + -3a2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-1 + 2.333333333a + a2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2.333333333a + 1 + a2 = 0 + 1

Reorder the terms:
-1 + 1 + 2.333333333a + a2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2.333333333a + a2 = 0 + 1
2.333333333a + a2 = 0 + 1

Combine like terms: 0 + 1 = 1
2.333333333a + a2 = 1

The a term is 2.333333333a.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333a + 1.361111112 + a2 = 1 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333a + a2 = 1 + 1.361111112

Combine like terms: 1 + 1.361111112 = 2.361111112
1.361111112 + 2.333333333a + a2 = 2.361111112

Factor a perfect square on the left side:
(a + 1.166666667)(a + 1.166666667) = 2.361111112

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(a + 1.166666667) equal to 1.536590743 and -1.536590743.

Subproblem 1
a + 1.166666667 = 1.536590743

Simplifying
a + 1.166666667 = 1.536590743

Reorder the terms:
1.166666667 + a = 1.536590743

Solving
1.166666667 + a = 1.536590743

Solving for variable 'a'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + a = 1.536590743 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + a = 1.536590743 + -1.166666667
a = 1.536590743 + -1.166666667

Combine like terms: 1.536590743 + -1.166666667 = 0.369924076
a = 0.369924076

Simplifying
a = 0.369924076

Subproblem 2
a + 1.166666667 = -1.536590743

Simplifying
a + 1.166666667 = -1.536590743

Reorder the terms:
1.166666667 + a = -1.536590743

Solving
1.166666667 + a = -1.536590743

Solving for variable 'a'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + a = -1.536590743 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + a = -1.536590743 + -1.166666667
a = -1.536590743 + -1.166666667

Combine like terms: -1.536590743 + -1.166666667 = -2.70325741
a = -2.70325741

Simplifying
a = -2.70325741

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.369924076, -2.70325741}