4x^2+24x+1=0

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Solution for 4x^2+24x+1=0 equation:


Simplifying
4x2 + 24x + 1 = 0

Reorder the terms:
1 + 24x + 4x2 = 0

Solving
1 + 24x + 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 + 6x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 6x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 6x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 6x + x2 = 0 + -0.25
6x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
6x + x2 = -0.25

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = -0.25 + 9

Reorder the terms:
9 + 6x + x2 = -0.25 + 9

Combine like terms: -0.25 + 9 = 8.75
9 + 6x + x2 = 8.75

Factor a perfect square on the left side:
(x + 3)(x + 3) = 8.75

Calculate the square root of the right side: 2.958039892

Break this problem into two subproblems by setting 
(x + 3) equal to 2.958039892 and -2.958039892.

Subproblem 1

x + 3 = 2.958039892 Simplifying x + 3 = 2.958039892 Reorder the terms: 3 + x = 2.958039892 Solving 3 + x = 2.958039892 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 2.958039892 + -3 Combine like terms: 3 + -3 = 0 0 + x = 2.958039892 + -3 x = 2.958039892 + -3 Combine like terms: 2.958039892 + -3 = -0.041960108 x = -0.041960108 Simplifying x = -0.041960108

Subproblem 2

x + 3 = -2.958039892 Simplifying x + 3 = -2.958039892 Reorder the terms: 3 + x = -2.958039892 Solving 3 + x = -2.958039892 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -2.958039892 + -3 Combine like terms: 3 + -3 = 0 0 + x = -2.958039892 + -3 x = -2.958039892 + -3 Combine like terms: -2.958039892 + -3 = -5.958039892 x = -5.958039892 Simplifying x = -5.958039892

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.041960108, -5.958039892}

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