6x+1=x^2-1

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


Simplifying
6x + 1 = x2 + -1

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

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

Solving
1 + 6x = -1 + x2

Solving for variable 'x'.

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

Combine like terms: 1 + 1 = 2
2 + 6x + -1x2 = -1 + x2 + 1 + -1x2

Reorder the terms:
2 + 6x + -1x2 = -1 + 1 + x2 + -1x2

Combine like terms: -1 + 1 = 0
2 + 6x + -1x2 = 0 + x2 + -1x2
2 + 6x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
2 + 6x + -1x2 = 0

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

Divide each side by '-1'.
-2 + -6x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -6x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -6x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -6x + x2 = 0 + 2
-6x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-6x + x2 = 2

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 = 2 + 9

Reorder the terms:
9 + -6x + x2 = 2 + 9

Combine like terms: 2 + 9 = 11
9 + -6x + x2 = 11

Factor a perfect square on the left side:
(x + -3)(x + -3) = 11

Calculate the square root of the right side: 3.31662479

Break this problem into two subproblems by setting 
(x + -3) equal to 3.31662479 and -3.31662479.

Subproblem 1

x + -3 = 3.31662479 Simplifying x + -3 = 3.31662479 Reorder the terms: -3 + x = 3.31662479 Solving -3 + x = 3.31662479 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 = 3.31662479 + 3 Combine like terms: -3 + 3 = 0 0 + x = 3.31662479 + 3 x = 3.31662479 + 3 Combine like terms: 3.31662479 + 3 = 6.31662479 x = 6.31662479 Simplifying x = 6.31662479

Subproblem 2

x + -3 = -3.31662479 Simplifying x + -3 = -3.31662479 Reorder the terms: -3 + x = -3.31662479 Solving -3 + x = -3.31662479 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 = -3.31662479 + 3 Combine like terms: -3 + 3 = 0 0 + x = -3.31662479 + 3 x = -3.31662479 + 3 Combine like terms: -3.31662479 + 3 = -0.31662479 x = -0.31662479 Simplifying x = -0.31662479

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.31662479, -0.31662479}

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