5x^2-x=x^2-9x+31

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Solution for 5x^2-x=x^2-9x+31 equation:


Simplifying
5x2 + -1x = x2 + -9x + 31

Reorder the terms:
-1x + 5x2 = x2 + -9x + 31

Reorder the terms:
-1x + 5x2 = 31 + -9x + x2

Solving
-1x + 5x2 = 31 + -9x + x2

Solving for variable 'x'.

Reorder the terms:
-31 + -1x + 9x + 5x2 + -1x2 = 31 + -9x + x2 + -31 + 9x + -1x2

Combine like terms: -1x + 9x = 8x
-31 + 8x + 5x2 + -1x2 = 31 + -9x + x2 + -31 + 9x + -1x2

Combine like terms: 5x2 + -1x2 = 4x2
-31 + 8x + 4x2 = 31 + -9x + x2 + -31 + 9x + -1x2

Reorder the terms:
-31 + 8x + 4x2 = 31 + -31 + -9x + 9x + x2 + -1x2

Combine like terms: 31 + -31 = 0
-31 + 8x + 4x2 = 0 + -9x + 9x + x2 + -1x2
-31 + 8x + 4x2 = -9x + 9x + x2 + -1x2

Combine like terms: -9x + 9x = 0
-31 + 8x + 4x2 = 0 + x2 + -1x2
-31 + 8x + 4x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-31 + 8x + 4x2 = 0

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

Divide each side by '4'.
-7.75 + 2x + x2 = 0

Move the constant term to the right:

Add '7.75' to each side of the equation.
-7.75 + 2x + 7.75 + x2 = 0 + 7.75

Reorder the terms:
-7.75 + 7.75 + 2x + x2 = 0 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + 2x + x2 = 0 + 7.75
2x + x2 = 0 + 7.75

Combine like terms: 0 + 7.75 = 7.75
2x + x2 = 7.75

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 7.75 + 1

Reorder the terms:
1 + 2x + x2 = 7.75 + 1

Combine like terms: 7.75 + 1 = 8.75
1 + 2x + x2 = 8.75

Factor a perfect square on the left side:
(x + 1)(x + 1) = 8.75

Calculate the square root of the right side: 2.958039892

Break this problem into two subproblems by setting 
(x + 1) equal to 2.958039892 and -2.958039892.

Subproblem 1

x + 1 = 2.958039892 Simplifying x + 1 = 2.958039892 Reorder the terms: 1 + x = 2.958039892 Solving 1 + x = 2.958039892 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 2.958039892 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.958039892 + -1 x = 2.958039892 + -1 Combine like terms: 2.958039892 + -1 = 1.958039892 x = 1.958039892 Simplifying x = 1.958039892

Subproblem 2

x + 1 = -2.958039892 Simplifying x + 1 = -2.958039892 Reorder the terms: 1 + x = -2.958039892 Solving 1 + x = -2.958039892 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -2.958039892 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.958039892 + -1 x = -2.958039892 + -1 Combine like terms: -2.958039892 + -1 = -3.958039892 x = -3.958039892 Simplifying x = -3.958039892

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.958039892, -3.958039892}

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