4x^2+13x+5=0

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Solution for 4x^2+13x+5=0 equation:


Simplifying
4x2 + 13x + 5 = 0

Reorder the terms:
5 + 13x + 4x2 = 0

Solving
5 + 13x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
1.25 + 3.25x + x2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + 3.25x + -1.25 + x2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + 3.25x + x2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + 3.25x + x2 = 0 + -1.25
3.25x + x2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
3.25x + x2 = -1.25

The x term is 3.25x.  Take half its coefficient (1.625).
Square it (2.640625) and add it to both sides.

Add '2.640625' to each side of the equation.
3.25x + 2.640625 + x2 = -1.25 + 2.640625

Reorder the terms:
2.640625 + 3.25x + x2 = -1.25 + 2.640625

Combine like terms: -1.25 + 2.640625 = 1.390625
2.640625 + 3.25x + x2 = 1.390625

Factor a perfect square on the left side:
(x + 1.625)(x + 1.625) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(x + 1.625) equal to 1.179247642 and -1.179247642.

Subproblem 1

x + 1.625 = 1.179247642 Simplifying x + 1.625 = 1.179247642 Reorder the terms: 1.625 + x = 1.179247642 Solving 1.625 + x = 1.179247642 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + x = 1.179247642 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + x = 1.179247642 + -1.625 x = 1.179247642 + -1.625 Combine like terms: 1.179247642 + -1.625 = -0.445752358 x = -0.445752358 Simplifying x = -0.445752358

Subproblem 2

x + 1.625 = -1.179247642 Simplifying x + 1.625 = -1.179247642 Reorder the terms: 1.625 + x = -1.179247642 Solving 1.625 + x = -1.179247642 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + x = -1.179247642 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + x = -1.179247642 + -1.625 x = -1.179247642 + -1.625 Combine like terms: -1.179247642 + -1.625 = -2.804247642 x = -2.804247642 Simplifying x = -2.804247642

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.445752358, -2.804247642}

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