(9x-2)(5x+6)=x

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Solution for (9x-2)(5x+6)=x equation:


Simplifying
(9x + -2)(5x + 6) = x

Reorder the terms:
(-2 + 9x)(5x + 6) = x

Reorder the terms:
(-2 + 9x)(6 + 5x) = x

Multiply (-2 + 9x) * (6 + 5x)
(-2(6 + 5x) + 9x * (6 + 5x)) = x
((6 * -2 + 5x * -2) + 9x * (6 + 5x)) = x
((-12 + -10x) + 9x * (6 + 5x)) = x
(-12 + -10x + (6 * 9x + 5x * 9x)) = x
(-12 + -10x + (54x + 45x2)) = x

Combine like terms: -10x + 54x = 44x
(-12 + 44x + 45x2) = x

Solving
-12 + 44x + 45x2 = x

Solving for variable 'x'.

Reorder the terms:
-12 + 44x + -1x + 45x2 = x + -1x

Combine like terms: 44x + -1x = 43x
-12 + 43x + 45x2 = x + -1x

Combine like terms: x + -1x = 0
-12 + 43x + 45x2 = 0

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

Divide each side by '45'.
-0.2666666667 + 0.9555555556x + x2 = 0

Move the constant term to the right:

Add '0.2666666667' to each side of the equation.
-0.2666666667 + 0.9555555556x + 0.2666666667 + x2 = 0 + 0.2666666667

Reorder the terms:
-0.2666666667 + 0.2666666667 + 0.9555555556x + x2 = 0 + 0.2666666667

Combine like terms: -0.2666666667 + 0.2666666667 = 0.0000000000
0.0000000000 + 0.9555555556x + x2 = 0 + 0.2666666667
0.9555555556x + x2 = 0 + 0.2666666667

Combine like terms: 0 + 0.2666666667 = 0.2666666667
0.9555555556x + x2 = 0.2666666667

The x term is 0.9555555556x.  Take half its coefficient (0.4777777778).
Square it (0.2282716050) and add it to both sides.

Add '0.2282716050' to each side of the equation.
0.9555555556x + 0.2282716050 + x2 = 0.2666666667 + 0.2282716050

Reorder the terms:
0.2282716050 + 0.9555555556x + x2 = 0.2666666667 + 0.2282716050

Combine like terms: 0.2666666667 + 0.2282716050 = 0.4949382717
0.2282716050 + 0.9555555556x + x2 = 0.4949382717

Factor a perfect square on the left side:
(x + 0.4777777778)(x + 0.4777777778) = 0.4949382717

Calculate the square root of the right side: 0.703518494

Break this problem into two subproblems by setting 
(x + 0.4777777778) equal to 0.703518494 and -0.703518494.

Subproblem 1

x + 0.4777777778 = 0.703518494 Simplifying x + 0.4777777778 = 0.703518494 Reorder the terms: 0.4777777778 + x = 0.703518494 Solving 0.4777777778 + x = 0.703518494 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4777777778' to each side of the equation. 0.4777777778 + -0.4777777778 + x = 0.703518494 + -0.4777777778 Combine like terms: 0.4777777778 + -0.4777777778 = 0.0000000000 0.0000000000 + x = 0.703518494 + -0.4777777778 x = 0.703518494 + -0.4777777778 Combine like terms: 0.703518494 + -0.4777777778 = 0.2257407162 x = 0.2257407162 Simplifying x = 0.2257407162

Subproblem 2

x + 0.4777777778 = -0.703518494 Simplifying x + 0.4777777778 = -0.703518494 Reorder the terms: 0.4777777778 + x = -0.703518494 Solving 0.4777777778 + x = -0.703518494 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4777777778' to each side of the equation. 0.4777777778 + -0.4777777778 + x = -0.703518494 + -0.4777777778 Combine like terms: 0.4777777778 + -0.4777777778 = 0.0000000000 0.0000000000 + x = -0.703518494 + -0.4777777778 x = -0.703518494 + -0.4777777778 Combine like terms: -0.703518494 + -0.4777777778 = -1.1812962718 x = -1.1812962718 Simplifying x = -1.1812962718

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.2257407162, -1.1812962718}

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