x^2=19x+1

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Solution for x^2=19x+1 equation:


Simplifying
x2 = 19x + 1

Reorder the terms:
x2 = 1 + 19x

Solving
x2 = 1 + 19x

Solving for variable 'x'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The x term is -19x.  Take half its coefficient (-9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
-19x + 90.25 + x2 = 1 + 90.25

Reorder the terms:
90.25 + -19x + x2 = 1 + 90.25

Combine like terms: 1 + 90.25 = 91.25
90.25 + -19x + x2 = 91.25

Factor a perfect square on the left side:
(x + -9.5)(x + -9.5) = 91.25

Calculate the square root of the right side: 9.552486587

Break this problem into two subproblems by setting 
(x + -9.5) equal to 9.552486587 and -9.552486587.

Subproblem 1

x + -9.5 = 9.552486587 Simplifying x + -9.5 = 9.552486587 Reorder the terms: -9.5 + x = 9.552486587 Solving -9.5 + x = 9.552486587 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + x = 9.552486587 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x = 9.552486587 + 9.5 x = 9.552486587 + 9.5 Combine like terms: 9.552486587 + 9.5 = 19.052486587 x = 19.052486587 Simplifying x = 19.052486587

Subproblem 2

x + -9.5 = -9.552486587 Simplifying x + -9.5 = -9.552486587 Reorder the terms: -9.5 + x = -9.552486587 Solving -9.5 + x = -9.552486587 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + x = -9.552486587 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x = -9.552486587 + 9.5 x = -9.552486587 + 9.5 Combine like terms: -9.552486587 + 9.5 = -0.052486587 x = -0.052486587 Simplifying x = -0.052486587

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.052486587, -0.052486587}

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