21x^2+51x+21=0

Solution for 21x^2+51x+21=0 equation:

Simplifying
21x2 + 51x + 21 = 0

Reorder the terms:
21 + 51x + 21x2 = 0

Solving
21 + 51x + 21x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(7 + 17x + 7x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(7 + 17x + 7x2)' equal to zero and attempt to solve:

Simplifying
7 + 17x + 7x2 = 0

Solving
7 + 17x + 7x2 = 0

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

Divide each side by '7'.
1 + 2.428571429x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.428571429x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 2.428571429x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.428571429x + x2 = 0 + -1
2.428571429x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.428571429x + x2 = -1

The x term is 2.428571429x.  Take half its coefficient (1.214285715).
Square it (1.474489798) and add it to both sides.

Add '1.474489798' to each side of the equation.
2.428571429x + 1.474489798 + x2 = -1 + 1.474489798

Reorder the terms:
1.474489798 + 2.428571429x + x2 = -1 + 1.474489798

Combine like terms: -1 + 1.474489798 = 0.474489798
1.474489798 + 2.428571429x + x2 = 0.474489798

Factor a perfect square on the left side:
(x + 1.214285715)(x + 1.214285715) = 0.474489798

Calculate the square root of the right side: 0.688832199

Break this problem into two subproblems by setting 
(x + 1.214285715) equal to 0.688832199 and -0.688832199.

Subproblem 1
x + 1.214285715 = 0.688832199

Simplifying
x + 1.214285715 = 0.688832199

Reorder the terms:
1.214285715 + x = 0.688832199

Solving
1.214285715 + x = 0.688832199

Solving for variable 'x'.

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

Add '-1.214285715' to each side of the equation.
1.214285715 + -1.214285715 + x = 0.688832199 + -1.214285715

Combine like terms: 1.214285715 + -1.214285715 = 0.000000000
0.000000000 + x = 0.688832199 + -1.214285715
x = 0.688832199 + -1.214285715

Combine like terms: 0.688832199 + -1.214285715 = -0.525453516
x = -0.525453516

Simplifying
x = -0.525453516

Subproblem 2
x + 1.214285715 = -0.688832199

Simplifying
x + 1.214285715 = -0.688832199

Reorder the terms:
1.214285715 + x = -0.688832199

Solving
1.214285715 + x = -0.688832199

Solving for variable 'x'.

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

Add '-1.214285715' to each side of the equation.
1.214285715 + -1.214285715 + x = -0.688832199 + -1.214285715

Combine like terms: 1.214285715 + -1.214285715 = 0.000000000
0.000000000 + x = -0.688832199 + -1.214285715
x = -0.688832199 + -1.214285715

Combine like terms: -0.688832199 + -1.214285715 = -1.903117914
x = -1.903117914

Simplifying
x = -1.903117914

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.525453516, -1.903117914}
Solution
x = {-0.525453516, -1.903117914}