7x+1=7x^2-1

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Solution for 7x+1=7x^2-1 equation:


Simplifying
7x + 1 = 7x2 + -1

Reorder the terms:
1 + 7x = 7x2 + -1

Reorder the terms:
1 + 7x = -1 + 7x2

Solving
1 + 7x = -1 + 7x2

Solving for variable 'x'.

Reorder the terms:
1 + 1 + 7x + -7x2 = -1 + 7x2 + 1 + -7x2

Combine like terms: 1 + 1 = 2
2 + 7x + -7x2 = -1 + 7x2 + 1 + -7x2

Reorder the terms:
2 + 7x + -7x2 = -1 + 1 + 7x2 + -7x2

Combine like terms: -1 + 1 = 0
2 + 7x + -7x2 = 0 + 7x2 + -7x2
2 + 7x + -7x2 = 7x2 + -7x2

Combine like terms: 7x2 + -7x2 = 0
2 + 7x + -7x2 = 0

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
-0.2857142857 + -1x + x2 = 0

Move the constant term to the right:

Add '0.2857142857' to each side of the equation.
-0.2857142857 + -1x + 0.2857142857 + x2 = 0 + 0.2857142857

Reorder the terms:
-0.2857142857 + 0.2857142857 + -1x + x2 = 0 + 0.2857142857

Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000
0.0000000000 + -1x + x2 = 0 + 0.2857142857
-1x + x2 = 0 + 0.2857142857

Combine like terms: 0 + 0.2857142857 = 0.2857142857
-1x + x2 = 0.2857142857

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 = 0.2857142857 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 0.2857142857 + 0.25

Combine like terms: 0.2857142857 + 0.25 = 0.5357142857
0.25 + -1x + x2 = 0.5357142857

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 0.5357142857

Calculate the square root of the right side: 0.731925055

Break this problem into two subproblems by setting 
(x + -0.5) equal to 0.731925055 and -0.731925055.

Subproblem 1

x + -0.5 = 0.731925055 Simplifying x + -0.5 = 0.731925055 Reorder the terms: -0.5 + x = 0.731925055 Solving -0.5 + x = 0.731925055 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 = 0.731925055 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.731925055 + 0.5 x = 0.731925055 + 0.5 Combine like terms: 0.731925055 + 0.5 = 1.231925055 x = 1.231925055 Simplifying x = 1.231925055

Subproblem 2

x + -0.5 = -0.731925055 Simplifying x + -0.5 = -0.731925055 Reorder the terms: -0.5 + x = -0.731925055 Solving -0.5 + x = -0.731925055 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 = -0.731925055 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.731925055 + 0.5 x = -0.731925055 + 0.5 Combine like terms: -0.731925055 + 0.5 = -0.231925055 x = -0.231925055 Simplifying x = -0.231925055

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.231925055, -0.231925055}

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