10x-x^2+x=5

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Solution for 10x-x^2+x=5 equation:


Simplifying
10x + -1x2 + x = 5

Reorder the terms:
10x + x + -1x2 = 5

Combine like terms: 10x + x = 11x
11x + -1x2 = 5

Solving
11x + -1x2 = 5

Solving for variable 'x'.

Reorder the terms:
-5 + 11x + -1x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 11x + -1x2 = 0

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

Divide each side by '-1'.
5 + -11x + x2 = 0

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -11x + -5 + x2 = 0 + -5

Reorder the terms:
5 + -5 + -11x + x2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -11x + x2 = 0 + -5
-11x + x2 = 0 + -5

Combine like terms: 0 + -5 = -5
-11x + x2 = -5

The x term is -11x.  Take half its coefficient (-5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
-11x + 30.25 + x2 = -5 + 30.25

Reorder the terms:
30.25 + -11x + x2 = -5 + 30.25

Combine like terms: -5 + 30.25 = 25.25
30.25 + -11x + x2 = 25.25

Factor a perfect square on the left side:
(x + -5.5)(x + -5.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(x + -5.5) equal to 5.024937811 and -5.024937811.

Subproblem 1

x + -5.5 = 5.024937811 Simplifying x + -5.5 = 5.024937811 Reorder the terms: -5.5 + x = 5.024937811 Solving -5.5 + x = 5.024937811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + x = 5.024937811 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + x = 5.024937811 + 5.5 x = 5.024937811 + 5.5 Combine like terms: 5.024937811 + 5.5 = 10.524937811 x = 10.524937811 Simplifying x = 10.524937811

Subproblem 2

x + -5.5 = -5.024937811 Simplifying x + -5.5 = -5.024937811 Reorder the terms: -5.5 + x = -5.024937811 Solving -5.5 + x = -5.024937811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + x = -5.024937811 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + x = -5.024937811 + 5.5 x = -5.024937811 + 5.5 Combine like terms: -5.024937811 + 5.5 = 0.475062189 x = 0.475062189 Simplifying x = 0.475062189

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.524937811, 0.475062189}

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