3x^2-5x^2+10x+4=0

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Solution for 3x^2-5x^2+10x+4=0 equation: 


Simplifying
3x2 + -5x2 + 10x + 4 = 0

Reorder the terms:
4 + 10x + 3x2 + -5x2 = 0

Combine like terms: 3x2 + -5x2 = -2x2
4 + 10x + -2x2 = 0

Solving
4 + 10x + -2x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(2 + 5x + -1x2) = 0

Ignore the factor 2.

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

Simplifying
2 + 5x + -1x2 = 0

Solving
2 + 5x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-2 + -5x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -5x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -5x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -5x + x2 = 0 + 2
-5x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-5x + x2 = 2

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 2 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 2 + 6.25

Combine like terms: 2 + 6.25 = 8.25
6.25 + -5x + x2 = 8.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.872281323 and -2.872281323.


Subproblem 1
x + -2.5 = 2.872281323

Simplifying
x + -2.5 = 2.872281323

Reorder the terms:
-2.5 + x = 2.872281323

Solving
-2.5 + x = 2.872281323

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 2.872281323 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 2.872281323 + 2.5
x = 2.872281323 + 2.5

Combine like terms: 2.872281323 + 2.5 = 5.372281323
x = 5.372281323

Simplifying
x = 5.372281323


Subproblem 2
x + -2.5 = -2.872281323

Simplifying
x + -2.5 = -2.872281323

Reorder the terms:
-2.5 + x = -2.872281323

Solving
-2.5 + x = -2.872281323

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -2.872281323 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -2.872281323 + 2.5
x = -2.872281323 + 2.5

Combine like terms: -2.872281323 + 2.5 = -0.372281323
x = -0.372281323

Simplifying
x = -0.372281323


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.372281323, -0.372281323}
Solution
x = {5.372281323, -0.372281323}

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