(5x^2-3x+22)+(6x^2+6x-67)=0

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Solution for (5x^2-3x+22)+(6x^2+6x-67)=0 equation:


Simplifying
(5x2 + -3x + 22) + (6x2 + 6x + -67) = 0

Reorder the terms:
(22 + -3x + 5x2) + (6x2 + 6x + -67) = 0

Remove parenthesis around (22 + -3x + 5x2)
22 + -3x + 5x2 + (6x2 + 6x + -67) = 0

Reorder the terms:
22 + -3x + 5x2 + (-67 + 6x + 6x2) = 0

Remove parenthesis around (-67 + 6x + 6x2)
22 + -3x + 5x2 + -67 + 6x + 6x2 = 0

Reorder the terms:
22 + -67 + -3x + 6x + 5x2 + 6x2 = 0

Combine like terms: 22 + -67 = -45
-45 + -3x + 6x + 5x2 + 6x2 = 0

Combine like terms: -3x + 6x = 3x
-45 + 3x + 5x2 + 6x2 = 0

Combine like terms: 5x2 + 6x2 = 11x2
-45 + 3x + 11x2 = 0

Solving
-45 + 3x + 11x2 = 0

Solving for variable 'x'.

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

Divide each side by '11'.
-4.090909091 + 0.2727272727x + x2 = 0

Move the constant term to the right:

Add '4.090909091' to each side of the equation.
-4.090909091 + 0.2727272727x + 4.090909091 + x2 = 0 + 4.090909091

Reorder the terms:
-4.090909091 + 4.090909091 + 0.2727272727x + x2 = 0 + 4.090909091

Combine like terms: -4.090909091 + 4.090909091 = 0.000000000
0.000000000 + 0.2727272727x + x2 = 0 + 4.090909091
0.2727272727x + x2 = 0 + 4.090909091

Combine like terms: 0 + 4.090909091 = 4.090909091
0.2727272727x + x2 = 4.090909091

The x term is 0.2727272727x.  Take half its coefficient (0.1363636364).
Square it (0.01859504133) and add it to both sides.

Add '0.01859504133' to each side of the equation.
0.2727272727x + 0.01859504133 + x2 = 4.090909091 + 0.01859504133

Reorder the terms:
0.01859504133 + 0.2727272727x + x2 = 4.090909091 + 0.01859504133

Combine like terms: 4.090909091 + 0.01859504133 = 4.10950413233
0.01859504133 + 0.2727272727x + x2 = 4.10950413233

Factor a perfect square on the left side:
(x + 0.1363636364)(x + 0.1363636364) = 4.10950413233

Calculate the square root of the right side: 2.027191193

Break this problem into two subproblems by setting 
(x + 0.1363636364) equal to 2.027191193 and -2.027191193.

Subproblem 1

x + 0.1363636364 = 2.027191193 Simplifying x + 0.1363636364 = 2.027191193 Reorder the terms: 0.1363636364 + x = 2.027191193 Solving 0.1363636364 + x = 2.027191193 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + x = 2.027191193 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + x = 2.027191193 + -0.1363636364 x = 2.027191193 + -0.1363636364 Combine like terms: 2.027191193 + -0.1363636364 = 1.8908275566 x = 1.8908275566 Simplifying x = 1.8908275566

Subproblem 2

x + 0.1363636364 = -2.027191193 Simplifying x + 0.1363636364 = -2.027191193 Reorder the terms: 0.1363636364 + x = -2.027191193 Solving 0.1363636364 + x = -2.027191193 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + x = -2.027191193 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + x = -2.027191193 + -0.1363636364 x = -2.027191193 + -0.1363636364 Combine like terms: -2.027191193 + -0.1363636364 = -2.1635548294 x = -2.1635548294 Simplifying x = -2.1635548294

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.8908275566, -2.1635548294}

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