4x^2+20x+23=0

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Solution for 4x^2+20x+23=0 equation:


Simplifying
4x2 + 20x + 23 = 0

Reorder the terms:
23 + 20x + 4x2 = 0

Solving
23 + 20x + 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'.
5.75 + 5x + x2 = 0

Move the constant term to the right:

Add '-5.75' to each side of the equation.
5.75 + 5x + -5.75 + x2 = 0 + -5.75

Reorder the terms:
5.75 + -5.75 + 5x + x2 = 0 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + 5x + x2 = 0 + -5.75
5x + x2 = 0 + -5.75

Combine like terms: 0 + -5.75 = -5.75
5x + x2 = -5.75

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = -5.75 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -5.75 + 6.25

Combine like terms: -5.75 + 6.25 = 0.5
6.25 + 5x + x2 = 0.5

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + 2.5) equal to 0.707106781 and -0.707106781.

Subproblem 1

x + 2.5 = 0.707106781 Simplifying x + 2.5 = 0.707106781 Reorder the terms: 2.5 + x = 0.707106781 Solving 2.5 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 0.707106781 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 0.707106781 + -2.5 x = 0.707106781 + -2.5 Combine like terms: 0.707106781 + -2.5 = -1.792893219 x = -1.792893219 Simplifying x = -1.792893219

Subproblem 2

x + 2.5 = -0.707106781 Simplifying x + 2.5 = -0.707106781 Reorder the terms: 2.5 + x = -0.707106781 Solving 2.5 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -0.707106781 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -0.707106781 + -2.5 x = -0.707106781 + -2.5 Combine like terms: -0.707106781 + -2.5 = -3.207106781 x = -3.207106781 Simplifying x = -3.207106781

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.792893219, -3.207106781}

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