3x+22=x^2+6

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Solution for 3x+22=x^2+6 equation:


Simplifying
3x + 22 = x2 + 6

Reorder the terms:
22 + 3x = x2 + 6

Reorder the terms:
22 + 3x = 6 + x2

Solving
22 + 3x = 6 + x2

Solving for variable 'x'.

Reorder the terms:
22 + -6 + 3x + -1x2 = 6 + x2 + -6 + -1x2

Combine like terms: 22 + -6 = 16
16 + 3x + -1x2 = 6 + x2 + -6 + -1x2

Reorder the terms:
16 + 3x + -1x2 = 6 + -6 + x2 + -1x2

Combine like terms: 6 + -6 = 0
16 + 3x + -1x2 = 0 + x2 + -1x2
16 + 3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
16 + 3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-16 + -3x + x2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + -3x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + -3x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + -3x + x2 = 0 + 16
-3x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
-3x + x2 = 16

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 16 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 16 + 2.25

Combine like terms: 16 + 2.25 = 18.25
2.25 + -3x + x2 = 18.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(x + -1.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

x + -1.5 = 4.272001873 Simplifying x + -1.5 = 4.272001873 Reorder the terms: -1.5 + x = 4.272001873 Solving -1.5 + x = 4.272001873 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 4.272001873 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 4.272001873 + 1.5 x = 4.272001873 + 1.5 Combine like terms: 4.272001873 + 1.5 = 5.772001873 x = 5.772001873 Simplifying x = 5.772001873

Subproblem 2

x + -1.5 = -4.272001873 Simplifying x + -1.5 = -4.272001873 Reorder the terms: -1.5 + x = -4.272001873 Solving -1.5 + x = -4.272001873 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -4.272001873 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -4.272001873 + 1.5 x = -4.272001873 + 1.5 Combine like terms: -4.272001873 + 1.5 = -2.772001873 x = -2.772001873 Simplifying x = -2.772001873

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.772001873, -2.772001873}

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