-6x=9x^2+4x-3

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Solution for -6x=9x^2+4x-3 equation:


Simplifying
-6x = 9x2 + 4x + -3

Reorder the terms:
-6x = -3 + 4x + 9x2

Solving
-6x = -3 + 4x + 9x2

Solving for variable 'x'.

Reorder the terms:
3 + -6x + -4x + -9x2 = -3 + 4x + 9x2 + 3 + -4x + -9x2

Combine like terms: -6x + -4x = -10x
3 + -10x + -9x2 = -3 + 4x + 9x2 + 3 + -4x + -9x2

Reorder the terms:
3 + -10x + -9x2 = -3 + 3 + 4x + -4x + 9x2 + -9x2

Combine like terms: -3 + 3 = 0
3 + -10x + -9x2 = 0 + 4x + -4x + 9x2 + -9x2
3 + -10x + -9x2 = 4x + -4x + 9x2 + -9x2

Combine like terms: 4x + -4x = 0
3 + -10x + -9x2 = 0 + 9x2 + -9x2
3 + -10x + -9x2 = 9x2 + -9x2

Combine like terms: 9x2 + -9x2 = 0
3 + -10x + -9x2 = 0

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

Divide each side by '-9'.
-0.3333333333 + 1.111111111x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 1.111111111x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 1.111111111x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 1.111111111x + x2 = 0 + 0.3333333333
1.111111111x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
1.111111111x + x2 = 0.3333333333

The x term is 1.111111111x.  Take half its coefficient (0.5555555555).
Square it (0.3086419752) and add it to both sides.

Add '0.3086419752' to each side of the equation.
1.111111111x + 0.3086419752 + x2 = 0.3333333333 + 0.3086419752

Reorder the terms:
0.3086419752 + 1.111111111x + x2 = 0.3333333333 + 0.3086419752

Combine like terms: 0.3333333333 + 0.3086419752 = 0.6419753085
0.3086419752 + 1.111111111x + x2 = 0.6419753085

Factor a perfect square on the left side:
(x + 0.5555555555)(x + 0.5555555555) = 0.6419753085

Calculate the square root of the right side: 0.801233617

Break this problem into two subproblems by setting 
(x + 0.5555555555) equal to 0.801233617 and -0.801233617.

Subproblem 1

x + 0.5555555555 = 0.801233617 Simplifying x + 0.5555555555 = 0.801233617 Reorder the terms: 0.5555555555 + x = 0.801233617 Solving 0.5555555555 + x = 0.801233617 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + x = 0.801233617 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + x = 0.801233617 + -0.5555555555 x = 0.801233617 + -0.5555555555 Combine like terms: 0.801233617 + -0.5555555555 = 0.2456780615 x = 0.2456780615 Simplifying x = 0.2456780615

Subproblem 2

x + 0.5555555555 = -0.801233617 Simplifying x + 0.5555555555 = -0.801233617 Reorder the terms: 0.5555555555 + x = -0.801233617 Solving 0.5555555555 + x = -0.801233617 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + x = -0.801233617 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + x = -0.801233617 + -0.5555555555 x = -0.801233617 + -0.5555555555 Combine like terms: -0.801233617 + -0.5555555555 = -1.3567891725 x = -1.3567891725 Simplifying x = -1.3567891725

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.2456780615, -1.3567891725}

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