2x^2+21x+1=0

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


Simplifying
2x2 + 21x + 1 = 0

Reorder the terms:
1 + 21x + 2x2 = 0

Solving
1 + 21x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
0.5 + 10.5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 10.5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 10.5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 10.5x + x2 = 0 + -0.5
10.5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
10.5x + x2 = -0.5

The x term is 10.5x.  Take half its coefficient (5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
10.5x + 27.5625 + x2 = -0.5 + 27.5625

Reorder the terms:
27.5625 + 10.5x + x2 = -0.5 + 27.5625

Combine like terms: -0.5 + 27.5625 = 27.0625
27.5625 + 10.5x + x2 = 27.0625

Factor a perfect square on the left side:
(x + 5.25)(x + 5.25) = 27.0625

Calculate the square root of the right side: 5.202163012

Break this problem into two subproblems by setting 
(x + 5.25) equal to 5.202163012 and -5.202163012.

Subproblem 1

x + 5.25 = 5.202163012 Simplifying x + 5.25 = 5.202163012 Reorder the terms: 5.25 + x = 5.202163012 Solving 5.25 + x = 5.202163012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + x = 5.202163012 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + x = 5.202163012 + -5.25 x = 5.202163012 + -5.25 Combine like terms: 5.202163012 + -5.25 = -0.047836988 x = -0.047836988 Simplifying x = -0.047836988

Subproblem 2

x + 5.25 = -5.202163012 Simplifying x + 5.25 = -5.202163012 Reorder the terms: 5.25 + x = -5.202163012 Solving 5.25 + x = -5.202163012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + x = -5.202163012 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + x = -5.202163012 + -5.25 x = -5.202163012 + -5.25 Combine like terms: -5.202163012 + -5.25 = -10.452163012 x = -10.452163012 Simplifying x = -10.452163012

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.047836988, -10.452163012}

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