4x^2+18x+10=0

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Solution for 4x^2+18x+10=0 equation: 


Simplifying
4x2 + 18x + 10 = 0

Reorder the terms:
10 + 18x + 4x2 = 0

Solving
10 + 18x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + 9x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(5 + 9x + 2x2)' equal to zero and attempt to solve:

Simplifying
5 + 9x + 2x2 = 0

Solving
5 + 9x + 2x2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
2.5 + 4.5x + x2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 4.5x + -2.5 + x2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 4.5x + x2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 4.5x + x2 = 0 + -2.5
4.5x + x2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
4.5x + x2 = -2.5

The x term is 4.5x.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5x + 5.0625 + x2 = -2.5 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = -2.5 + 5.0625

Combine like terms: -2.5 + 5.0625 = 2.5625
5.0625 + 4.5x + x2 = 2.5625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(x + 2.25) equal to 1.600781059 and -1.600781059.


Subproblem 1
x + 2.25 = 1.600781059

Simplifying
x + 2.25 = 1.600781059

Reorder the terms:
2.25 + x = 1.600781059

Solving
2.25 + x = 1.600781059

Solving for variable 'x'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + x = 1.600781059 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = 1.600781059 + -2.25
x = 1.600781059 + -2.25

Combine like terms: 1.600781059 + -2.25 = -0.649218941
x = -0.649218941

Simplifying
x = -0.649218941


Subproblem 2
x + 2.25 = -1.600781059

Simplifying
x + 2.25 = -1.600781059

Reorder the terms:
2.25 + x = -1.600781059

Solving
2.25 + x = -1.600781059

Solving for variable 'x'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + x = -1.600781059 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = -1.600781059 + -2.25
x = -1.600781059 + -2.25

Combine like terms: -1.600781059 + -2.25 = -3.850781059
x = -3.850781059

Simplifying
x = -3.850781059


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.649218941, -3.850781059}
Solution
x = {-0.649218941, -3.850781059}

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