4x^2+42x+54=110

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Solution for 4x^2+42x+54=110 equation:


Simplifying
4x2 + 42x + 54 = 110

Reorder the terms:
54 + 42x + 4x2 = 110

Solving
54 + 42x + 4x2 = 110

Solving for variable 'x'.

Reorder the terms:
54 + -110 + 42x + 4x2 = 110 + -110

Combine like terms: 54 + -110 = -56
-56 + 42x + 4x2 = 110 + -110

Combine like terms: 110 + -110 = 0
-56 + 42x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-28 + 21x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-28 + 21x + 2x2)' equal to zero and attempt to solve:

Simplifying
-28 + 21x + 2x2 = 0

Solving
-28 + 21x + 2x2 = 0

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

Divide each side by '2'.
-14 + 10.5x + x2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 10.5x + 14 + x2 = 0 + 14

Reorder the terms:
-14 + 14 + 10.5x + x2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 10.5x + x2 = 0 + 14
10.5x + x2 = 0 + 14

Combine like terms: 0 + 14 = 14
10.5x + x2 = 14

The x term is 10.5x.  Take half its coefficient (5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
10.5x + 27.5625 + x2 = 14 + 27.5625

Reorder the terms:
27.5625 + 10.5x + x2 = 14 + 27.5625

Combine like terms: 14 + 27.5625 = 41.5625
27.5625 + 10.5x + x2 = 41.5625

Factor a perfect square on the left side:
(x + 5.25)(x + 5.25) = 41.5625

Calculate the square root of the right side: 6.446898479

Break this problem into two subproblems by setting 
(x + 5.25) equal to 6.446898479 and -6.446898479.


Subproblem 1
x + 5.25 = 6.446898479

Simplifying
x + 5.25 = 6.446898479

Reorder the terms:
5.25 + x = 6.446898479

Solving
5.25 + x = 6.446898479

Solving for variable 'x'.

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

Add '-5.25' to each side of the equation.
5.25 + -5.25 + x = 6.446898479 + -5.25

Combine like terms: 5.25 + -5.25 = 0.00
0.00 + x = 6.446898479 + -5.25
x = 6.446898479 + -5.25

Combine like terms: 6.446898479 + -5.25 = 1.196898479
x = 1.196898479

Simplifying
x = 1.196898479


Subproblem 2
x + 5.25 = -6.446898479

Simplifying
x + 5.25 = -6.446898479

Reorder the terms:
5.25 + x = -6.446898479

Solving
5.25 + x = -6.446898479

Solving for variable 'x'.

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

Add '-5.25' to each side of the equation.
5.25 + -5.25 + x = -6.446898479 + -5.25

Combine like terms: 5.25 + -5.25 = 0.00
0.00 + x = -6.446898479 + -5.25
x = -6.446898479 + -5.25

Combine like terms: -6.446898479 + -5.25 = -11.696898479
x = -11.696898479

Simplifying
x = -11.696898479


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.196898479, -11.696898479}
Solution
x = {1.196898479, -11.696898479}

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