4x^2+21x+1=0

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Solution for 4x^2+21x+1=0 equation: 


Simplifying
4x2 + 21x + 1 = 0

Reorder the terms:
1 + 21x + 4x2 = 0

Solving
1 + 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'.
0.25 + 5.25x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 5.25x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 5.25x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 5.25x + x2 = 0 + -0.25
5.25x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
5.25x + x2 = -0.25

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 = -0.25 + 6.890625

Reorder the terms:
6.890625 + 5.25x + x2 = -0.25 + 6.890625

Combine like terms: -0.25 + 6.890625 = 6.640625
6.890625 + 5.25x + x2 = 6.640625

Factor a perfect square on the left side:
(x + 2.625)(x + 2.625) = 6.640625

Calculate the square root of the right side: 2.576941016

Break this problem into two subproblems by setting 
(x + 2.625) equal to 2.576941016 and -2.576941016.


Subproblem 1
x + 2.625 = 2.576941016

Simplifying
x + 2.625 = 2.576941016

Reorder the terms:
2.625 + x = 2.576941016

Solving
2.625 + x = 2.576941016

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 = 2.576941016 + -2.625

Combine like terms: 2.625 + -2.625 = 0.000
0.000 + x = 2.576941016 + -2.625
x = 2.576941016 + -2.625

Combine like terms: 2.576941016 + -2.625 = -0.048058984
x = -0.048058984

Simplifying
x = -0.048058984


Subproblem 2
x + 2.625 = -2.576941016

Simplifying
x + 2.625 = -2.576941016

Reorder the terms:
2.625 + x = -2.576941016

Solving
2.625 + x = -2.576941016

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 = -2.576941016 + -2.625

Combine like terms: 2.625 + -2.625 = 0.000
0.000 + x = -2.576941016 + -2.625
x = -2.576941016 + -2.625

Combine like terms: -2.576941016 + -2.625 = -5.201941016
x = -5.201941016

Simplifying
x = -5.201941016


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.048058984, -5.201941016}

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