x+3=x^2-6x+12

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Solution for x+3=x^2-6x+12 equation:


Simplifying
x + 3 = x2 + -6x + 12

Reorder the terms:
3 + x = x2 + -6x + 12

Reorder the terms:
3 + x = 12 + -6x + x2

Solving
3 + x = 12 + -6x + x2

Solving for variable 'x'.

Reorder the terms:
3 + -12 + x + 6x + -1x2 = 12 + -6x + x2 + -12 + 6x + -1x2

Combine like terms: 3 + -12 = -9
-9 + x + 6x + -1x2 = 12 + -6x + x2 + -12 + 6x + -1x2

Combine like terms: x + 6x = 7x
-9 + 7x + -1x2 = 12 + -6x + x2 + -12 + 6x + -1x2

Reorder the terms:
-9 + 7x + -1x2 = 12 + -12 + -6x + 6x + x2 + -1x2

Combine like terms: 12 + -12 = 0
-9 + 7x + -1x2 = 0 + -6x + 6x + x2 + -1x2
-9 + 7x + -1x2 = -6x + 6x + x2 + -1x2

Combine like terms: -6x + 6x = 0
-9 + 7x + -1x2 = 0 + x2 + -1x2
-9 + 7x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-9 + 7x + -1x2 = 0

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

Divide each side by '-1'.
9 + -7x + x2 = 0

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -7x + -9 + x2 = 0 + -9

Reorder the terms:
9 + -9 + -7x + x2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -7x + x2 = 0 + -9
-7x + x2 = 0 + -9

Combine like terms: 0 + -9 = -9
-7x + x2 = -9

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -9 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -9 + 12.25

Combine like terms: -9 + 12.25 = 3.25
12.25 + -7x + x2 = 3.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(x + -3.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

x + -3.5 = 1.802775638 Simplifying x + -3.5 = 1.802775638 Reorder the terms: -3.5 + x = 1.802775638 Solving -3.5 + x = 1.802775638 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 1.802775638 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 1.802775638 + 3.5 x = 1.802775638 + 3.5 Combine like terms: 1.802775638 + 3.5 = 5.302775638 x = 5.302775638 Simplifying x = 5.302775638

Subproblem 2

x + -3.5 = -1.802775638 Simplifying x + -3.5 = -1.802775638 Reorder the terms: -3.5 + x = -1.802775638 Solving -3.5 + x = -1.802775638 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -1.802775638 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -1.802775638 + 3.5 x = -1.802775638 + 3.5 Combine like terms: -1.802775638 + 3.5 = 1.697224362 x = 1.697224362 Simplifying x = 1.697224362

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.302775638, 1.697224362}

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