4x^2+18x+20=52

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Solution for 4x^2+18x+20=52 equation: 


Simplifying
4x2 + 18x + 20 = 52

Reorder the terms:
20 + 18x + 4x2 = 52

Solving
20 + 18x + 4x2 = 52

Solving for variable 'x'.

Reorder the terms:
20 + -52 + 18x + 4x2 = 52 + -52

Combine like terms: 20 + -52 = -32
-32 + 18x + 4x2 = 52 + -52

Combine like terms: 52 + -52 = 0
-32 + 18x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-16 + 9x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-16 + 9x + 2x2)' equal to zero and attempt to solve:

Simplifying
-16 + 9x + 2x2 = 0

Solving
-16 + 9x + 2x2 = 0

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

Divide each side by '2'.
-8 + 4.5x + x2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 4.5x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + 4.5x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 4.5x + x2 = 0 + 8
4.5x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
4.5x + x2 = 8

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 = 8 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 8 + 5.0625

Combine like terms: 8 + 5.0625 = 13.0625
5.0625 + 4.5x + x2 = 13.0625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 13.0625

Calculate the square root of the right side: 3.614208074

Break this problem into two subproblems by setting 
(x + 2.25) equal to 3.614208074 and -3.614208074.


Subproblem 1
x + 2.25 = 3.614208074

Simplifying
x + 2.25 = 3.614208074

Reorder the terms:
2.25 + x = 3.614208074

Solving
2.25 + x = 3.614208074

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 = 3.614208074 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = 3.614208074 + -2.25
x = 3.614208074 + -2.25

Combine like terms: 3.614208074 + -2.25 = 1.364208074
x = 1.364208074

Simplifying
x = 1.364208074


Subproblem 2
x + 2.25 = -3.614208074

Simplifying
x + 2.25 = -3.614208074

Reorder the terms:
2.25 + x = -3.614208074

Solving
2.25 + x = -3.614208074

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 = -3.614208074 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = -3.614208074 + -2.25
x = -3.614208074 + -2.25

Combine like terms: -3.614208074 + -2.25 = -5.864208074
x = -5.864208074

Simplifying
x = -5.864208074


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.364208074, -5.864208074}
Solution
x = {1.364208074, -5.864208074}

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