2x^2=6x+1

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


Simplifying
2x2 = 6x + 1

Reorder the terms:
2x2 = 1 + 6x

Solving
2x2 = 1 + 6x

Solving for variable 'x'.

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

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

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

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

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

Divide each side by '2'.
-0.5 + -3x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -3x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -3x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -3x + x2 = 0 + 0.5
-3x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-3x + x2 = 0.5

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 0.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 0.5 + 2.25

Combine like terms: 0.5 + 2.25 = 2.75
2.25 + -3x + x2 = 2.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.658312395 and -1.658312395.

Subproblem 1

x + -1.5 = 1.658312395 Simplifying x + -1.5 = 1.658312395 Reorder the terms: -1.5 + x = 1.658312395 Solving -1.5 + x = 1.658312395 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 1.658312395 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.658312395 + 1.5 x = 1.658312395 + 1.5 Combine like terms: 1.658312395 + 1.5 = 3.158312395 x = 3.158312395 Simplifying x = 3.158312395

Subproblem 2

x + -1.5 = -1.658312395 Simplifying x + -1.5 = -1.658312395 Reorder the terms: -1.5 + x = -1.658312395 Solving -1.5 + x = -1.658312395 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -1.658312395 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.658312395 + 1.5 x = -1.658312395 + 1.5 Combine like terms: -1.658312395 + 1.5 = -0.158312395 x = -0.158312395 Simplifying x = -0.158312395

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.158312395, -0.158312395}

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