4x^2+19x+2=0

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Solution for 4x^2+19x+2=0 equation:


Simplifying
4x2 + 19x + 2 = 0

Reorder the terms:
2 + 19x + 4x2 = 0

Solving
2 + 19x + 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'.
0.5 + 4.75x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 4.75x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 4.75x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 4.75x + x2 = 0 + -0.5
4.75x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
4.75x + x2 = -0.5

The x term is 4.75x.  Take half its coefficient (2.375).
Square it (5.640625) and add it to both sides.

Add '5.640625' to each side of the equation.
4.75x + 5.640625 + x2 = -0.5 + 5.640625

Reorder the terms:
5.640625 + 4.75x + x2 = -0.5 + 5.640625

Combine like terms: -0.5 + 5.640625 = 5.140625
5.640625 + 4.75x + x2 = 5.140625

Factor a perfect square on the left side:
(x + 2.375)(x + 2.375) = 5.140625

Calculate the square root of the right side: 2.267294643

Break this problem into two subproblems by setting 
(x + 2.375) equal to 2.267294643 and -2.267294643.

Subproblem 1

x + 2.375 = 2.267294643 Simplifying x + 2.375 = 2.267294643 Reorder the terms: 2.375 + x = 2.267294643 Solving 2.375 + x = 2.267294643 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + x = 2.267294643 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + x = 2.267294643 + -2.375 x = 2.267294643 + -2.375 Combine like terms: 2.267294643 + -2.375 = -0.107705357 x = -0.107705357 Simplifying x = -0.107705357

Subproblem 2

x + 2.375 = -2.267294643 Simplifying x + 2.375 = -2.267294643 Reorder the terms: 2.375 + x = -2.267294643 Solving 2.375 + x = -2.267294643 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + x = -2.267294643 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + x = -2.267294643 + -2.375 x = -2.267294643 + -2.375 Combine like terms: -2.267294643 + -2.375 = -4.642294643 x = -4.642294643 Simplifying x = -4.642294643

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.107705357, -4.642294643}

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