(2x-5)=x^2-27x+31

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Solution for (2x-5)=x^2-27x+31 equation:


Simplifying
(2x + -5) = x2 + -27x + 31

Reorder the terms:
(-5 + 2x) = x2 + -27x + 31

Remove parenthesis around (-5 + 2x)
-5 + 2x = x2 + -27x + 31

Reorder the terms:
-5 + 2x = 31 + -27x + x2

Solving
-5 + 2x = 31 + -27x + x2

Solving for variable 'x'.

Reorder the terms:
-5 + -31 + 2x + 27x + -1x2 = 31 + -27x + x2 + -31 + 27x + -1x2

Combine like terms: -5 + -31 = -36
-36 + 2x + 27x + -1x2 = 31 + -27x + x2 + -31 + 27x + -1x2

Combine like terms: 2x + 27x = 29x
-36 + 29x + -1x2 = 31 + -27x + x2 + -31 + 27x + -1x2

Reorder the terms:
-36 + 29x + -1x2 = 31 + -31 + -27x + 27x + x2 + -1x2

Combine like terms: 31 + -31 = 0
-36 + 29x + -1x2 = 0 + -27x + 27x + x2 + -1x2
-36 + 29x + -1x2 = -27x + 27x + x2 + -1x2

Combine like terms: -27x + 27x = 0
-36 + 29x + -1x2 = 0 + x2 + -1x2
-36 + 29x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-36 + 29x + -1x2 = 0

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

Divide each side by '-1'.
36 + -29x + x2 = 0

Move the constant term to the right:

Add '-36' to each side of the equation.
36 + -29x + -36 + x2 = 0 + -36

Reorder the terms:
36 + -36 + -29x + x2 = 0 + -36

Combine like terms: 36 + -36 = 0
0 + -29x + x2 = 0 + -36
-29x + x2 = 0 + -36

Combine like terms: 0 + -36 = -36
-29x + x2 = -36

The x term is -29x.  Take half its coefficient (-14.5).
Square it (210.25) and add it to both sides.

Add '210.25' to each side of the equation.
-29x + 210.25 + x2 = -36 + 210.25

Reorder the terms:
210.25 + -29x + x2 = -36 + 210.25

Combine like terms: -36 + 210.25 = 174.25
210.25 + -29x + x2 = 174.25

Factor a perfect square on the left side:
(x + -14.5)(x + -14.5) = 174.25

Calculate the square root of the right side: 13.200378782

Break this problem into two subproblems by setting 
(x + -14.5) equal to 13.200378782 and -13.200378782.

Subproblem 1

x + -14.5 = 13.200378782 Simplifying x + -14.5 = 13.200378782 Reorder the terms: -14.5 + x = 13.200378782 Solving -14.5 + x = 13.200378782 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '14.5' to each side of the equation. -14.5 + 14.5 + x = 13.200378782 + 14.5 Combine like terms: -14.5 + 14.5 = 0.0 0.0 + x = 13.200378782 + 14.5 x = 13.200378782 + 14.5 Combine like terms: 13.200378782 + 14.5 = 27.700378782 x = 27.700378782 Simplifying x = 27.700378782

Subproblem 2

x + -14.5 = -13.200378782 Simplifying x + -14.5 = -13.200378782 Reorder the terms: -14.5 + x = -13.200378782 Solving -14.5 + x = -13.200378782 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '14.5' to each side of the equation. -14.5 + 14.5 + x = -13.200378782 + 14.5 Combine like terms: -14.5 + 14.5 = 0.0 0.0 + x = -13.200378782 + 14.5 x = -13.200378782 + 14.5 Combine like terms: -13.200378782 + 14.5 = 1.299621218 x = 1.299621218 Simplifying x = 1.299621218

Solution

The solution to the problem is based on the solutions from the subproblems. x = {27.700378782, 1.299621218}

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