3x^2+75x+24=0

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Solution for 3x^2+75x+24=0 equation: 


Simplifying
3x2 + 75x + 24 = 0

Reorder the terms:
24 + 75x + 3x2 = 0

Solving
24 + 75x + 3x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(8 + 25x + x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(8 + 25x + x2)' equal to zero and attempt to solve:

Simplifying
8 + 25x + x2 = 0

Solving
8 + 25x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 25x + -8 + x2 = 0 + -8

Reorder the terms:
8 + -8 + 25x + x2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 25x + x2 = 0 + -8
25x + x2 = 0 + -8

Combine like terms: 0 + -8 = -8
25x + x2 = -8

The x term is 25x.  Take half its coefficient (12.5).
Square it (156.25) and add it to both sides.

Add '156.25' to each side of the equation.
25x + 156.25 + x2 = -8 + 156.25

Reorder the terms:
156.25 + 25x + x2 = -8 + 156.25

Combine like terms: -8 + 156.25 = 148.25
156.25 + 25x + x2 = 148.25

Factor a perfect square on the left side:
(x + 12.5)(x + 12.5) = 148.25

Calculate the square root of the right side: 12.175795662

Break this problem into two subproblems by setting 
(x + 12.5) equal to 12.175795662 and -12.175795662.


Subproblem 1
x + 12.5 = 12.175795662

Simplifying
x + 12.5 = 12.175795662

Reorder the terms:
12.5 + x = 12.175795662

Solving
12.5 + x = 12.175795662

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-12.5' to each side of the equation.
12.5 + -12.5 + x = 12.175795662 + -12.5

Combine like terms: 12.5 + -12.5 = 0.0
0.0 + x = 12.175795662 + -12.5
x = 12.175795662 + -12.5

Combine like terms: 12.175795662 + -12.5 = -0.324204338
x = -0.324204338

Simplifying
x = -0.324204338


Subproblem 2
x + 12.5 = -12.175795662

Simplifying
x + 12.5 = -12.175795662

Reorder the terms:
12.5 + x = -12.175795662

Solving
12.5 + x = -12.175795662

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-12.5' to each side of the equation.
12.5 + -12.5 + x = -12.175795662 + -12.5

Combine like terms: 12.5 + -12.5 = 0.0
0.0 + x = -12.175795662 + -12.5
x = -12.175795662 + -12.5

Combine like terms: -12.175795662 + -12.5 = -24.675795662
x = -24.675795662

Simplifying
x = -24.675795662


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.324204338, -24.675795662}
Solution
x = {-0.324204338, -24.675795662}

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