4x^2+21x+3=0

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Solution for 4x^2+21x+3=0 equation: 


Simplifying
4x2 + 21x + 3 = 0

Reorder the terms:
3 + 21x + 4x2 = 0

Solving
3 + 21x + 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'.
0.75 + 5.25x + x2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 5.25x + -0.75 + x2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 5.25x + x2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 5.25x + x2 = 0 + -0.75
5.25x + x2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
5.25x + x2 = -0.75

The x term is 5.25x.  Take half its coefficient (2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
5.25x + 6.890625 + x2 = -0.75 + 6.890625

Reorder the terms:
6.890625 + 5.25x + x2 = -0.75 + 6.890625

Combine like terms: -0.75 + 6.890625 = 6.140625
6.890625 + 5.25x + x2 = 6.140625

Factor a perfect square on the left side:
(x + 2.625)(x + 2.625) = 6.140625

Calculate the square root of the right side: 2.47802845

Break this problem into two subproblems by setting 
(x + 2.625) equal to 2.47802845 and -2.47802845.


Subproblem 1
x + 2.625 = 2.47802845

Simplifying
x + 2.625 = 2.47802845

Reorder the terms:
2.625 + x = 2.47802845

Solving
2.625 + x = 2.47802845

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.625' to each side of the equation.
2.625 + -2.625 + x = 2.47802845 + -2.625

Combine like terms: 2.625 + -2.625 = 0.000
0.000 + x = 2.47802845 + -2.625
x = 2.47802845 + -2.625

Combine like terms: 2.47802845 + -2.625 = -0.14697155
x = -0.14697155

Simplifying
x = -0.14697155


Subproblem 2
x + 2.625 = -2.47802845

Simplifying
x + 2.625 = -2.47802845

Reorder the terms:
2.625 + x = -2.47802845

Solving
2.625 + x = -2.47802845

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.625' to each side of the equation.
2.625 + -2.625 + x = -2.47802845 + -2.625

Combine like terms: 2.625 + -2.625 = 0.000
0.000 + x = -2.47802845 + -2.625
x = -2.47802845 + -2.625

Combine like terms: -2.47802845 + -2.625 = -5.10302845
x = -5.10302845

Simplifying
x = -5.10302845


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.14697155, -5.10302845}

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