4x+110=x^2+x

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Solution for 4x+110=x^2+x equation:


Simplifying
4x + 110 = x2 + x

Reorder the terms:
110 + 4x = x2 + x

Reorder the terms:
110 + 4x = x + x2

Solving
110 + 4x = x + x2

Solving for variable 'x'.

Combine like terms: 4x + -1x = 3x
110 + 3x + -1x2 = x + x2 + -1x + -1x2

Reorder the terms:
110 + 3x + -1x2 = x + -1x + x2 + -1x2

Combine like terms: x + -1x = 0
110 + 3x + -1x2 = 0 + x2 + -1x2
110 + 3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
110 + 3x + -1x2 = 0

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

Divide each side by '-1'.
-110 + -3x + x2 = 0

Move the constant term to the right:

Add '110' to each side of the equation.
-110 + -3x + 110 + x2 = 0 + 110

Reorder the terms:
-110 + 110 + -3x + x2 = 0 + 110

Combine like terms: -110 + 110 = 0
0 + -3x + x2 = 0 + 110
-3x + x2 = 0 + 110

Combine like terms: 0 + 110 = 110
-3x + x2 = 110

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 = 110 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 110 + 2.25

Combine like terms: 110 + 2.25 = 112.25
2.25 + -3x + x2 = 112.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 112.25

Calculate the square root of the right side: 10.59481005

Break this problem into two subproblems by setting 
(x + -1.5) equal to 10.59481005 and -10.59481005.

Subproblem 1

x + -1.5 = 10.59481005 Simplifying x + -1.5 = 10.59481005 Reorder the terms: -1.5 + x = 10.59481005 Solving -1.5 + x = 10.59481005 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 = 10.59481005 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 10.59481005 + 1.5 x = 10.59481005 + 1.5 Combine like terms: 10.59481005 + 1.5 = 12.09481005 x = 12.09481005 Simplifying x = 12.09481005

Subproblem 2

x + -1.5 = -10.59481005 Simplifying x + -1.5 = -10.59481005 Reorder the terms: -1.5 + x = -10.59481005 Solving -1.5 + x = -10.59481005 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 = -10.59481005 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -10.59481005 + 1.5 x = -10.59481005 + 1.5 Combine like terms: -10.59481005 + 1.5 = -9.09481005 x = -9.09481005 Simplifying x = -9.09481005

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12.09481005, -9.09481005}

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