8.5-3x^2=3.5x+2x^2

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Solution for 8.5-3x^2=3.5x+2x^2 equation: 


Simplifying
8.5 + -3x2 = 3.5x + 2x2

Solving
8.5 + -3x2 = 3.5x + 2x2

Solving for variable 'x'.

Reorder the terms:
8.5 + -3.5x + -3x2 + -2x2 = 3.5x + 2x2 + -3.5x + -2x2

Combine like terms: -3x2 + -2x2 = -5x2
8.5 + -3.5x + -5x2 = 3.5x + 2x2 + -3.5x + -2x2

Reorder the terms:
8.5 + -3.5x + -5x2 = 3.5x + -3.5x + 2x2 + -2x2

Combine like terms: 3.5x + -3.5x = 0.0
8.5 + -3.5x + -5x2 = 0.0 + 2x2 + -2x2
8.5 + -3.5x + -5x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
8.5 + -3.5x + -5x2 = 0

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

Divide each side by '-5'.
-1.7 + 0.7x + x2 = 0

Move the constant term to the right:

Add '1.7' to each side of the equation.
-1.7 + 0.7x + 1.7 + x2 = 0 + 1.7

Reorder the terms:
-1.7 + 1.7 + 0.7x + x2 = 0 + 1.7

Combine like terms: -1.7 + 1.7 = 0.0
0.0 + 0.7x + x2 = 0 + 1.7
0.7x + x2 = 0 + 1.7

Combine like terms: 0 + 1.7 = 1.7
0.7x + x2 = 1.7

The x term is 0.7x.  Take half its coefficient (0.35).
Square it (0.1225) and add it to both sides.

Add '0.1225' to each side of the equation.
0.7x + 0.1225 + x2 = 1.7 + 0.1225

Reorder the terms:
0.1225 + 0.7x + x2 = 1.7 + 0.1225

Combine like terms: 1.7 + 0.1225 = 1.8225
0.1225 + 0.7x + x2 = 1.8225

Factor a perfect square on the left side:
(x + 0.35)(x + 0.35) = 1.8225

Calculate the square root of the right side: 1.35

Break this problem into two subproblems by setting 
(x + 0.35) equal to 1.35 and -1.35.


Subproblem 1
x + 0.35 = 1.35

Simplifying
x + 0.35 = 1.35

Reorder the terms:
0.35 + x = 1.35

Solving
0.35 + x = 1.35

Solving for variable 'x'.

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

Add '-0.35' to each side of the equation.
0.35 + -0.35 + x = 1.35 + -0.35

Combine like terms: 0.35 + -0.35 = 0.00
0.00 + x = 1.35 + -0.35
x = 1.35 + -0.35

Combine like terms: 1.35 + -0.35 = 1
x = 1

Simplifying
x = 1


Subproblem 2
x + 0.35 = -1.35

Simplifying
x + 0.35 = -1.35

Reorder the terms:
0.35 + x = -1.35

Solving
0.35 + x = -1.35

Solving for variable 'x'.

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

Add '-0.35' to each side of the equation.
0.35 + -0.35 + x = -1.35 + -0.35

Combine like terms: 0.35 + -0.35 = 0.00
0.00 + x = -1.35 + -0.35
x = -1.35 + -0.35

Combine like terms: -1.35 + -0.35 = -1.7
x = -1.7

Simplifying
x = -1.7


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1, -1.7}

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