4x^2+24x+5=0

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Solution for 4x^2+24x+5=0 equation:


Simplifying
4x2 + 24x + 5 = 0

Reorder the terms:
5 + 24x + 4x2 = 0

Solving
5 + 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'.
1.25 + 6x + x2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + 6x + -1.25 + x2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + 6x + x2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + 6x + x2 = 0 + -1.25
6x + x2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
6x + x2 = -1.25

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 = -1.25 + 9

Reorder the terms:
9 + 6x + x2 = -1.25 + 9

Combine like terms: -1.25 + 9 = 7.75
9 + 6x + x2 = 7.75

Factor a perfect square on the left side:
(x + 3)(x + 3) = 7.75

Calculate the square root of the right side: 2.783882181

Break this problem into two subproblems by setting 
(x + 3) equal to 2.783882181 and -2.783882181.

Subproblem 1

x + 3 = 2.783882181 Simplifying x + 3 = 2.783882181 Reorder the terms: 3 + x = 2.783882181 Solving 3 + x = 2.783882181 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 = 2.783882181 + -3 Combine like terms: 3 + -3 = 0 0 + x = 2.783882181 + -3 x = 2.783882181 + -3 Combine like terms: 2.783882181 + -3 = -0.216117819 x = -0.216117819 Simplifying x = -0.216117819

Subproblem 2

x + 3 = -2.783882181 Simplifying x + 3 = -2.783882181 Reorder the terms: 3 + x = -2.783882181 Solving 3 + x = -2.783882181 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 = -2.783882181 + -3 Combine like terms: 3 + -3 = 0 0 + x = -2.783882181 + -3 x = -2.783882181 + -3 Combine like terms: -2.783882181 + -3 = -5.783882181 x = -5.783882181 Simplifying x = -5.783882181

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.216117819, -5.783882181}

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