3x^2-x^2-27x+9=0

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Solution for 3x^2-x^2-27x+9=0 equation:


Simplifying
3x2 + -1x2 + -27x + 9 = 0

Reorder the terms:
9 + -27x + 3x2 + -1x2 = 0

Combine like terms: 3x2 + -1x2 = 2x2
9 + -27x + 2x2 = 0

Solving
9 + -27x + 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 + -13.5x + x2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + -13.5x + -4.5 + x2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + -13.5x + x2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + -13.5x + x2 = 0 + -4.5
-13.5x + x2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
-13.5x + x2 = -4.5

The x term is -13.5x.  Take half its coefficient (-6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
-13.5x + 45.5625 + x2 = -4.5 + 45.5625

Reorder the terms:
45.5625 + -13.5x + x2 = -4.5 + 45.5625

Combine like terms: -4.5 + 45.5625 = 41.0625
45.5625 + -13.5x + x2 = 41.0625

Factor a perfect square on the left side:
(x + -6.75)(x + -6.75) = 41.0625

Calculate the square root of the right side: 6.408002809

Break this problem into two subproblems by setting 
(x + -6.75) equal to 6.408002809 and -6.408002809.

Subproblem 1

x + -6.75 = 6.408002809 Simplifying x + -6.75 = 6.408002809 Reorder the terms: -6.75 + x = 6.408002809 Solving -6.75 + x = 6.408002809 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.75' to each side of the equation. -6.75 + 6.75 + x = 6.408002809 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + x = 6.408002809 + 6.75 x = 6.408002809 + 6.75 Combine like terms: 6.408002809 + 6.75 = 13.158002809 x = 13.158002809 Simplifying x = 13.158002809

Subproblem 2

x + -6.75 = -6.408002809 Simplifying x + -6.75 = -6.408002809 Reorder the terms: -6.75 + x = -6.408002809 Solving -6.75 + x = -6.408002809 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.75' to each side of the equation. -6.75 + 6.75 + x = -6.408002809 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + x = -6.408002809 + 6.75 x = -6.408002809 + 6.75 Combine like terms: -6.408002809 + 6.75 = 0.341997191 x = 0.341997191 Simplifying x = 0.341997191

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13.158002809, 0.341997191}

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