32x-3x^2=19

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Solution for 32x-3x^2=19 equation: 


Simplifying
32x + -3x2 = 19

Solving
32x + -3x2 = 19

Solving for variable 'x'.

Reorder the terms:
-19 + 32x + -3x2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + 32x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
6.333333333 + -10.66666667x + x2 = 0

Move the constant term to the right:

Add '-6.333333333' to each side of the equation.
6.333333333 + -10.66666667x + -6.333333333 + x2 = 0 + -6.333333333

Reorder the terms:
6.333333333 + -6.333333333 + -10.66666667x + x2 = 0 + -6.333333333

Combine like terms: 6.333333333 + -6.333333333 = 0.000000000
0.000000000 + -10.66666667x + x2 = 0 + -6.333333333
-10.66666667x + x2 = 0 + -6.333333333

Combine like terms: 0 + -6.333333333 = -6.333333333
-10.66666667x + x2 = -6.333333333

The x term is -10.66666667x.  Take half its coefficient (-5.333333335).
Square it (28.44444446) and add it to both sides.

Add '28.44444446' to each side of the equation.
-10.66666667x + 28.44444446 + x2 = -6.333333333 + 28.44444446

Reorder the terms:
28.44444446 + -10.66666667x + x2 = -6.333333333 + 28.44444446

Combine like terms: -6.333333333 + 28.44444446 = 22.111111127
28.44444446 + -10.66666667x + x2 = 22.111111127

Factor a perfect square on the left side:
(x + -5.333333335)(x + -5.333333335) = 22.111111127

Calculate the square root of the right side: 4.702245328

Break this problem into two subproblems by setting 
(x + -5.333333335) equal to 4.702245328 and -4.702245328.


Subproblem 1
x + -5.333333335 = 4.702245328

Simplifying
x + -5.333333335 = 4.702245328

Reorder the terms:
-5.333333335 + x = 4.702245328

Solving
-5.333333335 + x = 4.702245328

Solving for variable 'x'.

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

Add '5.333333335' to each side of the equation.
-5.333333335 + 5.333333335 + x = 4.702245328 + 5.333333335

Combine like terms: -5.333333335 + 5.333333335 = 0.000000000
0.000000000 + x = 4.702245328 + 5.333333335
x = 4.702245328 + 5.333333335

Combine like terms: 4.702245328 + 5.333333335 = 10.035578663
x = 10.035578663

Simplifying
x = 10.035578663


Subproblem 2
x + -5.333333335 = -4.702245328

Simplifying
x + -5.333333335 = -4.702245328

Reorder the terms:
-5.333333335 + x = -4.702245328

Solving
-5.333333335 + x = -4.702245328

Solving for variable 'x'.

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

Add '5.333333335' to each side of the equation.
-5.333333335 + 5.333333335 + x = -4.702245328 + 5.333333335

Combine like terms: -5.333333335 + 5.333333335 = 0.000000000
0.000000000 + x = -4.702245328 + 5.333333335
x = -4.702245328 + 5.333333335

Combine like terms: -4.702245328 + 5.333333335 = 0.631088007
x = 0.631088007

Simplifying
x = 0.631088007


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {10.035578663, 0.631088007}

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