2x^2+1=x+5

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Solution for 2x^2+1=x+5 equation:


Simplifying
2x2 + 1 = x + 5

Reorder the terms:
1 + 2x2 = x + 5

Reorder the terms:
1 + 2x2 = 5 + x

Solving
1 + 2x2 = 5 + x

Solving for variable 'x'.

Reorder the terms:
1 + -5 + -1x + 2x2 = 5 + x + -5 + -1x

Combine like terms: 1 + -5 = -4
-4 + -1x + 2x2 = 5 + x + -5 + -1x

Reorder the terms:
-4 + -1x + 2x2 = 5 + -5 + x + -1x

Combine like terms: 5 + -5 = 0
-4 + -1x + 2x2 = 0 + x + -1x
-4 + -1x + 2x2 = x + -1x

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

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

Divide each side by '2'.
-2 + -0.5x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -0.5x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -0.5x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -0.5x + x2 = 0 + 2
-0.5x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-0.5x + x2 = 2

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 2 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 2 + 0.0625

Combine like terms: 2 + 0.0625 = 2.0625
0.0625 + -0.5x + x2 = 2.0625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.436140662 and -1.436140662.

Subproblem 1

x + -0.25 = 1.436140662 Simplifying x + -0.25 = 1.436140662 Reorder the terms: -0.25 + x = 1.436140662 Solving -0.25 + x = 1.436140662 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 1.436140662 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.436140662 + 0.25 x = 1.436140662 + 0.25 Combine like terms: 1.436140662 + 0.25 = 1.686140662 x = 1.686140662 Simplifying x = 1.686140662

Subproblem 2

x + -0.25 = -1.436140662 Simplifying x + -0.25 = -1.436140662 Reorder the terms: -0.25 + x = -1.436140662 Solving -0.25 + x = -1.436140662 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -1.436140662 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.436140662 + 0.25 x = -1.436140662 + 0.25 Combine like terms: -1.436140662 + 0.25 = -1.186140662 x = -1.186140662 Simplifying x = -1.186140662

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.686140662, -1.186140662}

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