x^2-19=x

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Solution for x^2-19=x equation:


Simplifying
x2 + -19 = x

Reorder the terms:
-19 + x2 = x

Solving
-19 + x2 = x

Solving for variable 'x'.

Reorder the terms:
-19 + -1x + x2 = x + -1x

Combine like terms: x + -1x = 0
-19 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + -1x + 19 + x2 = 0 + 19

Reorder the terms:
-19 + 19 + -1x + x2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + -1x + x2 = 0 + 19
-1x + x2 = 0 + 19

Combine like terms: 0 + 19 = 19
-1x + x2 = 19

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 = 19 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 19 + 0.25

Combine like terms: 19 + 0.25 = 19.25
0.25 + -1x + x2 = 19.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 19.25

Calculate the square root of the right side: 4.387482194

Break this problem into two subproblems by setting 
(x + -0.5) equal to 4.387482194 and -4.387482194.

Subproblem 1

x + -0.5 = 4.387482194 Simplifying x + -0.5 = 4.387482194 Reorder the terms: -0.5 + x = 4.387482194 Solving -0.5 + x = 4.387482194 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 = 4.387482194 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 4.387482194 + 0.5 x = 4.387482194 + 0.5 Combine like terms: 4.387482194 + 0.5 = 4.887482194 x = 4.887482194 Simplifying x = 4.887482194

Subproblem 2

x + -0.5 = -4.387482194 Simplifying x + -0.5 = -4.387482194 Reorder the terms: -0.5 + x = -4.387482194 Solving -0.5 + x = -4.387482194 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 = -4.387482194 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -4.387482194 + 0.5 x = -4.387482194 + 0.5 Combine like terms: -4.387482194 + 0.5 = -3.887482194 x = -3.887482194 Simplifying x = -3.887482194

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.887482194, -3.887482194}

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