0=-3x^2+6x+7

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Solution for 0=-3x^2+6x+7 equation:


Simplifying
0 = -3x2 + 6x + 7

Reorder the terms:
0 = 7 + 6x + -3x2

Solving
0 = 7 + 6x + -3x2

Solving for variable 'x'.

Combine like terms: 0 + -7 = -7
-7 + -6x + 3x2 = 7 + 6x + -3x2 + -7 + -6x + 3x2

Reorder the terms:
-7 + -6x + 3x2 = 7 + -7 + 6x + -6x + -3x2 + 3x2

Combine like terms: 7 + -7 = 0
-7 + -6x + 3x2 = 0 + 6x + -6x + -3x2 + 3x2
-7 + -6x + 3x2 = 6x + -6x + -3x2 + 3x2

Combine like terms: 6x + -6x = 0
-7 + -6x + 3x2 = 0 + -3x2 + 3x2
-7 + -6x + 3x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
-7 + -6x + 3x2 = 0

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

Divide each side by '3'.
-2.333333333 + -2x + x2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -2x + 2.333333333 + x2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -2x + x2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -2x + x2 = 0 + 2.333333333
-2x + x2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-2x + x2 = 2.333333333

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 2.333333333 + 1

Reorder the terms:
1 + -2x + x2 = 2.333333333 + 1

Combine like terms: 2.333333333 + 1 = 3.333333333
1 + -2x + x2 = 3.333333333

Factor a perfect square on the left side:
(x + -1)(x + -1) = 3.333333333

Calculate the square root of the right side: 1.825741858

Break this problem into two subproblems by setting 
(x + -1) equal to 1.825741858 and -1.825741858.


Subproblem 1
x + -1 = 1.825741858

Simplifying
x + -1 = 1.825741858

Reorder the terms:
-1 + x = 1.825741858

Solving
-1 + x = 1.825741858

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = 1.825741858 + 1

Combine like terms: -1 + 1 = 0
0 + x = 1.825741858 + 1
x = 1.825741858 + 1

Combine like terms: 1.825741858 + 1 = 2.825741858
x = 2.825741858

Simplifying
x = 2.825741858


Subproblem 2
x + -1 = -1.825741858

Simplifying
x + -1 = -1.825741858

Reorder the terms:
-1 + x = -1.825741858

Solving
-1 + x = -1.825741858

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = -1.825741858 + 1

Combine like terms: -1 + 1 = 0
0 + x = -1.825741858 + 1
x = -1.825741858 + 1

Combine like terms: -1.825741858 + 1 = -0.825741858
x = -0.825741858

Simplifying
x = -0.825741858


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.825741858, -0.825741858}

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