x^2=x+38

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Solution for x^2=x+38 equation:


Simplifying
x2 = x + 38

Reorder the terms:
x2 = 38 + x

Solving
x2 = 38 + x

Solving for variable 'x'.

Reorder the terms:
-38 + -1x + x2 = 38 + x + -38 + -1x

Reorder the terms:
-38 + -1x + x2 = 38 + -38 + x + -1x

Combine like terms: 38 + -38 = 0
-38 + -1x + x2 = 0 + x + -1x
-38 + -1x + x2 = x + -1x

Combine like terms: x + -1x = 0
-38 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '38' to each side of the equation.
-38 + -1x + 38 + x2 = 0 + 38

Reorder the terms:
-38 + 38 + -1x + x2 = 0 + 38

Combine like terms: -38 + 38 = 0
0 + -1x + x2 = 0 + 38
-1x + x2 = 0 + 38

Combine like terms: 0 + 38 = 38
-1x + x2 = 38

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 38 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 38 + 0.25

Combine like terms: 38 + 0.25 = 38.25
0.25 + -1x + x2 = 38.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 38.25

Calculate the square root of the right side: 6.184658438

Break this problem into two subproblems by setting 
(x + -0.5) equal to 6.184658438 and -6.184658438.

Subproblem 1

x + -0.5 = 6.184658438 Simplifying x + -0.5 = 6.184658438 Reorder the terms: -0.5 + x = 6.184658438 Solving -0.5 + x = 6.184658438 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 6.184658438 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 6.184658438 + 0.5 x = 6.184658438 + 0.5 Combine like terms: 6.184658438 + 0.5 = 6.684658438 x = 6.684658438 Simplifying x = 6.684658438

Subproblem 2

x + -0.5 = -6.184658438 Simplifying x + -0.5 = -6.184658438 Reorder the terms: -0.5 + x = -6.184658438 Solving -0.5 + x = -6.184658438 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -6.184658438 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -6.184658438 + 0.5 x = -6.184658438 + 0.5 Combine like terms: -6.184658438 + 0.5 = -5.684658438 x = -5.684658438 Simplifying x = -5.684658438

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.684658438, -5.684658438}

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