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 = 2.0625
1.5625 + -2.5x + x2 = 2.0625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(x + -1.25) equal to 1.436140662 and -1.436140662.

Subproblem 1

x + -1.25 = 1.436140662 Simplifying x + -1.25 = 1.436140662 Reorder the terms: -1.25 + x = 1.436140662 Solving -1.25 + x = 1.436140662 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 = 1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.436140662 + 1.25 x = 1.436140662 + 1.25 Combine like terms: 1.436140662 + 1.25 = 2.686140662 x = 2.686140662 Simplifying x = 2.686140662

Subproblem 2

x + -1.25 = -1.436140662 Simplifying x + -1.25 = -1.436140662 Reorder the terms: -1.25 + x = -1.436140662 Solving -1.25 + x = -1.436140662 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 = -1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.436140662 + 1.25 x = -1.436140662 + 1.25 Combine like terms: -1.436140662 + 1.25 = -0.186140662 x = -0.186140662 Simplifying x = -0.186140662

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.686140662, -0.186140662}

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