x^2+1=2x^2+4x-23

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Solution for x^2+1=2x^2+4x-23 equation:


Simplifying
x2 + 1 = 2x2 + 4x + -23

Reorder the terms:
1 + x2 = 2x2 + 4x + -23

Reorder the terms:
1 + x2 = -23 + 4x + 2x2

Solving
1 + x2 = -23 + 4x + 2x2

Solving for variable 'x'.

Reorder the terms:
1 + 23 + -4x + x2 + -2x2 = -23 + 4x + 2x2 + 23 + -4x + -2x2

Combine like terms: 1 + 23 = 24
24 + -4x + x2 + -2x2 = -23 + 4x + 2x2 + 23 + -4x + -2x2

Combine like terms: x2 + -2x2 = -1x2
24 + -4x + -1x2 = -23 + 4x + 2x2 + 23 + -4x + -2x2

Reorder the terms:
24 + -4x + -1x2 = -23 + 23 + 4x + -4x + 2x2 + -2x2

Combine like terms: -23 + 23 = 0
24 + -4x + -1x2 = 0 + 4x + -4x + 2x2 + -2x2
24 + -4x + -1x2 = 4x + -4x + 2x2 + -2x2

Combine like terms: 4x + -4x = 0
24 + -4x + -1x2 = 0 + 2x2 + -2x2
24 + -4x + -1x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
24 + -4x + -1x2 = 0

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

Divide each side by '-1'.
-24 + 4x + x2 = 0

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + 4x + 24 + x2 = 0 + 24

Reorder the terms:
-24 + 24 + 4x + x2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + 4x + x2 = 0 + 24
4x + x2 = 0 + 24

Combine like terms: 0 + 24 = 24
4x + x2 = 24

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = 24 + 4

Reorder the terms:
4 + 4x + x2 = 24 + 4

Combine like terms: 24 + 4 = 28
4 + 4x + x2 = 28

Factor a perfect square on the left side:
(x + 2)(x + 2) = 28

Calculate the square root of the right side: 5.291502622

Break this problem into two subproblems by setting 
(x + 2) equal to 5.291502622 and -5.291502622.

Subproblem 1

x + 2 = 5.291502622 Simplifying x + 2 = 5.291502622 Reorder the terms: 2 + x = 5.291502622 Solving 2 + x = 5.291502622 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = 5.291502622 + -2 Combine like terms: 2 + -2 = 0 0 + x = 5.291502622 + -2 x = 5.291502622 + -2 Combine like terms: 5.291502622 + -2 = 3.291502622 x = 3.291502622 Simplifying x = 3.291502622

Subproblem 2

x + 2 = -5.291502622 Simplifying x + 2 = -5.291502622 Reorder the terms: 2 + x = -5.291502622 Solving 2 + x = -5.291502622 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = -5.291502622 + -2 Combine like terms: 2 + -2 = 0 0 + x = -5.291502622 + -2 x = -5.291502622 + -2 Combine like terms: -5.291502622 + -2 = -7.291502622 x = -7.291502622 Simplifying x = -7.291502622

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.291502622, -7.291502622}

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