x^2-6x+(x^2-9)=0

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Solution for x^2-6x+(x^2-9)=0 equation:


Simplifying
x2 + -6x + (x2 + -9) = 0

Reorder the terms:
x2 + -6x + (-9 + x2) = 0

Remove parenthesis around (-9 + x2)
x2 + -6x + -9 + x2 = 0

Reorder the terms:
-9 + -6x + x2 + x2 = 0

Combine like terms: x2 + x2 = 2x2
-9 + -6x + 2x2 = 0

Solving
-9 + -6x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-4.5 + -3x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -3x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -3x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -3x + x2 = 0 + 4.5
-3x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-3x + x2 = 4.5

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 = 4.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 4.5 + 2.25

Combine like terms: 4.5 + 2.25 = 6.75
2.25 + -3x + x2 = 6.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(x + -1.5) equal to 2.598076211 and -2.598076211.

Subproblem 1

x + -1.5 = 2.598076211 Simplifying x + -1.5 = 2.598076211 Reorder the terms: -1.5 + x = 2.598076211 Solving -1.5 + x = 2.598076211 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 = 2.598076211 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 2.598076211 + 1.5 x = 2.598076211 + 1.5 Combine like terms: 2.598076211 + 1.5 = 4.098076211 x = 4.098076211 Simplifying x = 4.098076211

Subproblem 2

x + -1.5 = -2.598076211 Simplifying x + -1.5 = -2.598076211 Reorder the terms: -1.5 + x = -2.598076211 Solving -1.5 + x = -2.598076211 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 = -2.598076211 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -2.598076211 + 1.5 x = -2.598076211 + 1.5 Combine like terms: -2.598076211 + 1.5 = -1.098076211 x = -1.098076211 Simplifying x = -1.098076211

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.098076211, -1.098076211}

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