3x^2+2x-2.25=0

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Solution for 3x^2+2x-2.25=0 equation: 


Simplifying
3x2 + 2x + -2.25 = 0

Reorder the terms:
-2.25 + 2x + 3x2 = 0

Solving
-2.25 + 2x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-0.75 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 0.6666666667x + 0.75 + x2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 0.6666666667x + x2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 0.6666666667x + x2 = 0 + 0.75
0.6666666667x + x2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
0.6666666667x + x2 = 0.75

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 0.75 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 0.75 + 0.1111111112

Combine like terms: 0.75 + 0.1111111112 = 0.8611111112
0.1111111112 + 0.6666666667x + x2 = 0.8611111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 0.8611111112

Calculate the square root of the right side: 0.927960727

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 0.927960727 and -0.927960727.


Subproblem 1
x + 0.3333333334 = 0.927960727

Simplifying
x + 0.3333333334 = 0.927960727

Reorder the terms:
0.3333333334 + x = 0.927960727

Solving
0.3333333334 + x = 0.927960727

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = 0.927960727 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = 0.927960727 + -0.3333333334
x = 0.927960727 + -0.3333333334

Combine like terms: 0.927960727 + -0.3333333334 = 0.5946273936
x = 0.5946273936

Simplifying
x = 0.5946273936


Subproblem 2
x + 0.3333333334 = -0.927960727

Simplifying
x + 0.3333333334 = -0.927960727

Reorder the terms:
0.3333333334 + x = -0.927960727

Solving
0.3333333334 + x = -0.927960727

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = -0.927960727 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = -0.927960727 + -0.3333333334
x = -0.927960727 + -0.3333333334

Combine like terms: -0.927960727 + -0.3333333334 = -1.2612940604
x = -1.2612940604

Simplifying
x = -1.2612940604


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.5946273936, -1.2612940604}

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