6x=x^2+1

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Solution for 6x=x^2+1 equation:


Simplifying
6x = x2 + 1

Reorder the terms:
6x = 1 + x2

Solving
6x = 1 + x2

Solving for variable 'x'.

Reorder the terms:
-1 + 6x + -1x2 = 1 + x2 + -1 + -1x2

Reorder the terms:
-1 + 6x + -1x2 = 1 + -1 + x2 + -1x2

Combine like terms: 1 + -1 = 0
-1 + 6x + -1x2 = 0 + x2 + -1x2
-1 + 6x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-1 + 6x + -1x2 = 0

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

Divide each side by '-1'.
1 + -6x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -6x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -6x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -6x + x2 = 0 + -1
-6x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-6x + x2 = -1

The x term is -6x.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x + 9 + x2 = -1 + 9

Reorder the terms:
9 + -6x + x2 = -1 + 9

Combine like terms: -1 + 9 = 8
9 + -6x + x2 = 8

Factor a perfect square on the left side:
(x + -3)(x + -3) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + -3) equal to 2.828427125 and -2.828427125.

Subproblem 1

x + -3 = 2.828427125 Simplifying x + -3 = 2.828427125 Reorder the terms: -3 + x = 2.828427125 Solving -3 + x = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + x = 2.828427125 + 3 x = 2.828427125 + 3 Combine like terms: 2.828427125 + 3 = 5.828427125 x = 5.828427125 Simplifying x = 5.828427125

Subproblem 2

x + -3 = -2.828427125 Simplifying x + -3 = -2.828427125 Reorder the terms: -3 + x = -2.828427125 Solving -3 + x = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = -2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + x = -2.828427125 + 3 x = -2.828427125 + 3 Combine like terms: -2.828427125 + 3 = 0.171572875 x = 0.171572875 Simplifying x = 0.171572875

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.828427125, 0.171572875}

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