2x^2+13x+9=0

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Solution for 2x^2+13x+9=0 equation:


Simplifying
2x2 + 13x + 9 = 0

Reorder the terms:
9 + 13x + 2x2 = 0

Solving
9 + 13x + 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'.
4.5 + 6.5x + x2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + 6.5x + -4.5 + x2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + 6.5x + x2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + 6.5x + x2 = 0 + -4.5
6.5x + x2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
6.5x + x2 = -4.5

The x term is 6.5x.  Take half its coefficient (3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
6.5x + 10.5625 + x2 = -4.5 + 10.5625

Reorder the terms:
10.5625 + 6.5x + x2 = -4.5 + 10.5625

Combine like terms: -4.5 + 10.5625 = 6.0625
10.5625 + 6.5x + x2 = 6.0625

Factor a perfect square on the left side:
(x + 3.25)(x + 3.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(x + 3.25) equal to 2.46221445 and -2.46221445.

Subproblem 1

x + 3.25 = 2.46221445 Simplifying x + 3.25 = 2.46221445 Reorder the terms: 3.25 + x = 2.46221445 Solving 3.25 + x = 2.46221445 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + x = 2.46221445 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + x = 2.46221445 + -3.25 x = 2.46221445 + -3.25 Combine like terms: 2.46221445 + -3.25 = -0.78778555 x = -0.78778555 Simplifying x = -0.78778555

Subproblem 2

x + 3.25 = -2.46221445 Simplifying x + 3.25 = -2.46221445 Reorder the terms: 3.25 + x = -2.46221445 Solving 3.25 + x = -2.46221445 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + x = -2.46221445 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + x = -2.46221445 + -3.25 x = -2.46221445 + -3.25 Combine like terms: -2.46221445 + -3.25 = -5.71221445 x = -5.71221445 Simplifying x = -5.71221445

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.78778555, -5.71221445}

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