2x^2+9x+3=0

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Solution for 2x^2+9x+3=0 equation:


Simplifying
2x2 + 9x + 3 = 0

Reorder the terms:
3 + 9x + 2x2 = 0

Solving
3 + 9x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
1.5 + 4.5x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 4.5x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 4.5x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 4.5x + x2 = 0 + -1.5
4.5x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
4.5x + x2 = -1.5

The x term is 4.5x.  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.
4.5x + 5.0625 + x2 = -1.5 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = -1.5 + 5.0625

Combine like terms: -1.5 + 5.0625 = 3.5625
5.0625 + 4.5x + x2 = 3.5625

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

Calculate the square root of the right side: 1.887458609

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

Subproblem 1

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

Subproblem 2

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

Solution

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

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