5a-2a^2=1

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Solution for 5a-2a^2=1 equation:


Simplifying
5a + -2a2 = 1

Solving
5a + -2a2 = 1

Solving for variable 'a'.

Reorder the terms:
-1 + 5a + -2a2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 5a + -2a2 = 0

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

Divide each side by '-2'.
0.5 + -2.5a + a2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2.5a + -0.5 + a2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2.5a + a2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2.5a + a2 = 0 + -0.5
-2.5a + a2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2.5a + a2 = -0.5

The a term is -2.5a.  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.
-2.5a + 1.5625 + a2 = -0.5 + 1.5625

Reorder the terms:
1.5625 + -2.5a + a2 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + -2.5a + a2 = 1.0625

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

Calculate the square root of the right side: 1.030776406

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

Subproblem 1

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

Subproblem 2

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

Solution

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

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