7x=x^2-1

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


Simplifying
7x = x2 + -1

Reorder the terms:
7x = -1 + x2

Solving
7x = -1 + x2

Solving for variable 'x'.

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

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

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

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

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

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

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -7x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -7x + x2 = 0 + 1

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

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

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = 1 + 12.25

Reorder the terms:
12.25 + -7x + x2 = 1 + 12.25

Combine like terms: 1 + 12.25 = 13.25
12.25 + -7x + x2 = 13.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(x + -3.5) equal to 3.640054945 and -3.640054945.

Subproblem 1

x + -3.5 = 3.640054945 Simplifying x + -3.5 = 3.640054945 Reorder the terms: -3.5 + x = 3.640054945 Solving -3.5 + x = 3.640054945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 3.640054945 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 3.640054945 + 3.5 x = 3.640054945 + 3.5 Combine like terms: 3.640054945 + 3.5 = 7.140054945 x = 7.140054945 Simplifying x = 7.140054945

Subproblem 2

x + -3.5 = -3.640054945 Simplifying x + -3.5 = -3.640054945 Reorder the terms: -3.5 + x = -3.640054945 Solving -3.5 + x = -3.640054945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -3.640054945 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -3.640054945 + 3.5 x = -3.640054945 + 3.5 Combine like terms: -3.640054945 + 3.5 = -0.140054945 x = -0.140054945 Simplifying x = -0.140054945

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.140054945, -0.140054945}

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