2a=4a^2-1

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Solution for 2a=4a^2-1 equation:


Simplifying
2a = 4a2 + -1

Reorder the terms:
2a = -1 + 4a2

Solving
2a = -1 + 4a2

Solving for variable 'a'.

Reorder the terms:
1 + 2a + -4a2 = -1 + 4a2 + 1 + -4a2

Reorder the terms:
1 + 2a + -4a2 = -1 + 1 + 4a2 + -4a2

Combine like terms: -1 + 1 = 0
1 + 2a + -4a2 = 0 + 4a2 + -4a2
1 + 2a + -4a2 = 4a2 + -4a2

Combine like terms: 4a2 + -4a2 = 0
1 + 2a + -4a2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-0.25 + -0.5a + a2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -0.5a + 0.25 + a2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -0.5a + a2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -0.5a + a2 = 0 + 0.25
-0.5a + a2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-0.5a + a2 = 0.25

The a term is -0.5a.  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.
-0.5a + 0.0625 + a2 = 0.25 + 0.0625

Reorder the terms:
0.0625 + -0.5a + a2 = 0.25 + 0.0625

Combine like terms: 0.25 + 0.0625 = 0.3125
0.0625 + -0.5a + a2 = 0.3125

Factor a perfect square on the left side:
(a + -0.25)(a + -0.25) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(a + -0.25) equal to 0.559016994 and -0.559016994.

Subproblem 1

a + -0.25 = 0.559016994 Simplifying a + -0.25 = 0.559016994 Reorder the terms: -0.25 + a = 0.559016994 Solving -0.25 + a = 0.559016994 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = 0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 0.559016994 + 0.25 a = 0.559016994 + 0.25 Combine like terms: 0.559016994 + 0.25 = 0.809016994 a = 0.809016994 Simplifying a = 0.809016994

Subproblem 2

a + -0.25 = -0.559016994 Simplifying a + -0.25 = -0.559016994 Reorder the terms: -0.25 + a = -0.559016994 Solving -0.25 + a = -0.559016994 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = -0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -0.559016994 + 0.25 a = -0.559016994 + 0.25 Combine like terms: -0.559016994 + 0.25 = -0.309016994 a = -0.309016994 Simplifying a = -0.309016994

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.809016994, -0.309016994}

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