5x=2x^2+1

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


Simplifying
5x = 2x2 + 1

Reorder the terms:
5x = 1 + 2x2

Solving
5x = 1 + 2x2

Solving for variable 'x'.

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

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

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

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

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

Divide each side by '-2'.
0.5 + -2.5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2.5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2.5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2.5x + x2 = 0 + -0.5
-2.5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2.5x + x2 = -0.5

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = -0.5 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + -2.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + -1.25) equal to 1.030776406 and -1.030776406.

Subproblem 1

x + -1.25 = 1.030776406 Simplifying x + -1.25 = 1.030776406 Reorder the terms: -1.25 + x = 1.030776406 Solving -1.25 + x = 1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.030776406 + 1.25 x = 1.030776406 + 1.25 Combine like terms: 1.030776406 + 1.25 = 2.280776406 x = 2.280776406 Simplifying x = 2.280776406

Subproblem 2

x + -1.25 = -1.030776406 Simplifying x + -1.25 = -1.030776406 Reorder the terms: -1.25 + x = -1.030776406 Solving -1.25 + x = -1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.030776406 + 1.25 x = -1.030776406 + 1.25 Combine like terms: -1.030776406 + 1.25 = 0.219223594 x = 0.219223594 Simplifying x = 0.219223594

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.280776406, 0.219223594}

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