12x^2-3x^2+5x-22=0

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Solution for 12x^2-3x^2+5x-22=0 equation:


Simplifying
12x2 + -3x2 + 5x + -22 = 0

Reorder the terms:
-22 + 5x + 12x2 + -3x2 = 0

Combine like terms: 12x2 + -3x2 = 9x2
-22 + 5x + 9x2 = 0

Solving
-22 + 5x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-2.444444444 + 0.5555555556x + x2 = 0

Move the constant term to the right:

Add '2.444444444' to each side of the equation.
-2.444444444 + 0.5555555556x + 2.444444444 + x2 = 0 + 2.444444444

Reorder the terms:
-2.444444444 + 2.444444444 + 0.5555555556x + x2 = 0 + 2.444444444

Combine like terms: -2.444444444 + 2.444444444 = 0.000000000
0.000000000 + 0.5555555556x + x2 = 0 + 2.444444444
0.5555555556x + x2 = 0 + 2.444444444

Combine like terms: 0 + 2.444444444 = 2.444444444
0.5555555556x + x2 = 2.444444444

The x term is 0.5555555556x.  Take half its coefficient (0.2777777778).
Square it (0.07716049384) and add it to both sides.

Add '0.07716049384' to each side of the equation.
0.5555555556x + 0.07716049384 + x2 = 2.444444444 + 0.07716049384

Reorder the terms:
0.07716049384 + 0.5555555556x + x2 = 2.444444444 + 0.07716049384

Combine like terms: 2.444444444 + 0.07716049384 = 2.52160493784
0.07716049384 + 0.5555555556x + x2 = 2.52160493784

Factor a perfect square on the left side:
(x + 0.2777777778)(x + 0.2777777778) = 2.52160493784

Calculate the square root of the right side: 1.587956214

Break this problem into two subproblems by setting 
(x + 0.2777777778) equal to 1.587956214 and -1.587956214.

Subproblem 1

x + 0.2777777778 = 1.587956214 Simplifying x + 0.2777777778 = 1.587956214 Reorder the terms: 0.2777777778 + x = 1.587956214 Solving 0.2777777778 + x = 1.587956214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + x = 1.587956214 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + x = 1.587956214 + -0.2777777778 x = 1.587956214 + -0.2777777778 Combine like terms: 1.587956214 + -0.2777777778 = 1.3101784362 x = 1.3101784362 Simplifying x = 1.3101784362

Subproblem 2

x + 0.2777777778 = -1.587956214 Simplifying x + 0.2777777778 = -1.587956214 Reorder the terms: 0.2777777778 + x = -1.587956214 Solving 0.2777777778 + x = -1.587956214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + x = -1.587956214 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + x = -1.587956214 + -0.2777777778 x = -1.587956214 + -0.2777777778 Combine like terms: -1.587956214 + -0.2777777778 = -1.8657339918 x = -1.8657339918 Simplifying x = -1.8657339918

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3101784362, -1.8657339918}

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