4x^2+23x+1=0

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Solution for 4x^2+23x+1=0 equation:


Simplifying
4x2 + 23x + 1 = 0

Reorder the terms:
1 + 23x + 4x2 = 0

Solving
1 + 23x + 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 + 5.75x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 5.75x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 5.75x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 5.75x + x2 = 0 + -0.25
5.75x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
5.75x + x2 = -0.25

The x term is 5.75x.  Take half its coefficient (2.875).
Square it (8.265625) and add it to both sides.

Add '8.265625' to each side of the equation.
5.75x + 8.265625 + x2 = -0.25 + 8.265625

Reorder the terms:
8.265625 + 5.75x + x2 = -0.25 + 8.265625

Combine like terms: -0.25 + 8.265625 = 8.015625
8.265625 + 5.75x + x2 = 8.015625

Factor a perfect square on the left side:
(x + 2.875)(x + 2.875) = 8.015625

Calculate the square root of the right side: 2.831187913

Break this problem into two subproblems by setting 
(x + 2.875) equal to 2.831187913 and -2.831187913.

Subproblem 1

x + 2.875 = 2.831187913 Simplifying x + 2.875 = 2.831187913 Reorder the terms: 2.875 + x = 2.831187913 Solving 2.875 + x = 2.831187913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + x = 2.831187913 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + x = 2.831187913 + -2.875 x = 2.831187913 + -2.875 Combine like terms: 2.831187913 + -2.875 = -0.043812087 x = -0.043812087 Simplifying x = -0.043812087

Subproblem 2

x + 2.875 = -2.831187913 Simplifying x + 2.875 = -2.831187913 Reorder the terms: 2.875 + x = -2.831187913 Solving 2.875 + x = -2.831187913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + x = -2.831187913 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + x = -2.831187913 + -2.875 x = -2.831187913 + -2.875 Combine like terms: -2.831187913 + -2.875 = -5.706187913 x = -5.706187913 Simplifying x = -5.706187913

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.043812087, -5.706187913}

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