10x^2+194x+8451=9293

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Solution for 10x^2+194x+8451=9293 equation: 


Simplifying
10x2 + 194x + 8451 = 9293

Reorder the terms:
8451 + 194x + 10x2 = 9293

Solving
8451 + 194x + 10x2 = 9293

Solving for variable 'x'.

Reorder the terms:
8451 + -9293 + 194x + 10x2 = 9293 + -9293

Combine like terms: 8451 + -9293 = -842
-842 + 194x + 10x2 = 9293 + -9293

Combine like terms: 9293 + -9293 = 0
-842 + 194x + 10x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-421 + 97x + 5x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-421 + 97x + 5x2)' equal to zero and attempt to solve:

Simplifying
-421 + 97x + 5x2 = 0

Solving
-421 + 97x + 5x2 = 0

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

Divide each side by '5'.
-84.2 + 19.4x + x2 = 0

Move the constant term to the right:

Add '84.2' to each side of the equation.
-84.2 + 19.4x + 84.2 + x2 = 0 + 84.2

Reorder the terms:
-84.2 + 84.2 + 19.4x + x2 = 0 + 84.2

Combine like terms: -84.2 + 84.2 = 0.0
0.0 + 19.4x + x2 = 0 + 84.2
19.4x + x2 = 0 + 84.2

Combine like terms: 0 + 84.2 = 84.2
19.4x + x2 = 84.2

The x term is 19.4x.  Take half its coefficient (9.7).
Square it (94.09) and add it to both sides.

Add '94.09' to each side of the equation.
19.4x + 94.09 + x2 = 84.2 + 94.09

Reorder the terms:
94.09 + 19.4x + x2 = 84.2 + 94.09

Combine like terms: 84.2 + 94.09 = 178.29
94.09 + 19.4x + x2 = 178.29

Factor a perfect square on the left side:
(x + 9.7)(x + 9.7) = 178.29

Calculate the square root of the right side: 13.352527851

Break this problem into two subproblems by setting 
(x + 9.7) equal to 13.352527851 and -13.352527851.


Subproblem 1
x + 9.7 = 13.352527851

Simplifying
x + 9.7 = 13.352527851

Reorder the terms:
9.7 + x = 13.352527851

Solving
9.7 + x = 13.352527851

Solving for variable 'x'.

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

Add '-9.7' to each side of the equation.
9.7 + -9.7 + x = 13.352527851 + -9.7

Combine like terms: 9.7 + -9.7 = 0.0
0.0 + x = 13.352527851 + -9.7
x = 13.352527851 + -9.7

Combine like terms: 13.352527851 + -9.7 = 3.652527851
x = 3.652527851

Simplifying
x = 3.652527851


Subproblem 2
x + 9.7 = -13.352527851

Simplifying
x + 9.7 = -13.352527851

Reorder the terms:
9.7 + x = -13.352527851

Solving
9.7 + x = -13.352527851

Solving for variable 'x'.

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

Add '-9.7' to each side of the equation.
9.7 + -9.7 + x = -13.352527851 + -9.7

Combine like terms: 9.7 + -9.7 = 0.0
0.0 + x = -13.352527851 + -9.7
x = -13.352527851 + -9.7

Combine like terms: -13.352527851 + -9.7 = -23.052527851
x = -23.052527851

Simplifying
x = -23.052527851


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.652527851, -23.052527851}
Solution
x = {3.652527851, -23.052527851}

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