2a^2-9a+3=0

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Solution for 2a^2-9a+3=0 equation:


Simplifying
2a2 + -9a + 3 = 0

Reorder the terms:
3 + -9a + 2a2 = 0

Solving
3 + -9a + 2a2 = 0

Solving for variable 'a'.

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

Divide each side by '2'.
1.5 + -4.5a + a2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -4.5a + -1.5 + a2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -4.5a + a2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -4.5a + a2 = 0 + -1.5
-4.5a + a2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-4.5a + a2 = -1.5

The a term is -4.5a.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5a + 5.0625 + a2 = -1.5 + 5.0625

Reorder the terms:
5.0625 + -4.5a + a2 = -1.5 + 5.0625

Combine like terms: -1.5 + 5.0625 = 3.5625
5.0625 + -4.5a + a2 = 3.5625

Factor a perfect square on the left side:
(a + -2.25)(a + -2.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(a + -2.25) equal to 1.887458609 and -1.887458609.

Subproblem 1

a + -2.25 = 1.887458609 Simplifying a + -2.25 = 1.887458609 Reorder the terms: -2.25 + a = 1.887458609 Solving -2.25 + a = 1.887458609 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + a = 1.887458609 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = 1.887458609 + 2.25 a = 1.887458609 + 2.25 Combine like terms: 1.887458609 + 2.25 = 4.137458609 a = 4.137458609 Simplifying a = 4.137458609

Subproblem 2

a + -2.25 = -1.887458609 Simplifying a + -2.25 = -1.887458609 Reorder the terms: -2.25 + a = -1.887458609 Solving -2.25 + a = -1.887458609 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + a = -1.887458609 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = -1.887458609 + 2.25 a = -1.887458609 + 2.25 Combine like terms: -1.887458609 + 2.25 = 0.362541391 a = 0.362541391 Simplifying a = 0.362541391

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.137458609, 0.362541391}

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