0=-3x^2+4x+9

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Solution for 0=-3x^2+4x+9 equation:


Simplifying
0 = -3x2 + 4x + 9

Reorder the terms:
0 = 9 + 4x + -3x2

Solving
0 = 9 + 4x + -3x2

Solving for variable 'x'.

Combine like terms: 0 + -9 = -9
-9 + -4x + 3x2 = 9 + 4x + -3x2 + -9 + -4x + 3x2

Reorder the terms:
-9 + -4x + 3x2 = 9 + -9 + 4x + -4x + -3x2 + 3x2

Combine like terms: 9 + -9 = 0
-9 + -4x + 3x2 = 0 + 4x + -4x + -3x2 + 3x2
-9 + -4x + 3x2 = 4x + -4x + -3x2 + 3x2

Combine like terms: 4x + -4x = 0
-9 + -4x + 3x2 = 0 + -3x2 + 3x2
-9 + -4x + 3x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
-9 + -4x + 3x2 = 0

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

Divide each side by '3'.
-3 + -1.333333333x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -1.333333333x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + -1.333333333x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1.333333333x + x2 = 0 + 3
-1.333333333x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1.333333333x + x2 = 3

The x term is -1.333333333x.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333x + 0.4444444442 + x2 = 3 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333x + x2 = 3 + 0.4444444442

Combine like terms: 3 + 0.4444444442 = 3.4444444442
0.4444444442 + -1.333333333x + x2 = 3.4444444442

Factor a perfect square on the left side:
(x + -0.6666666665)(x + -0.6666666665) = 3.4444444442

Calculate the square root of the right side: 1.855921454

Break this problem into two subproblems by setting 
(x + -0.6666666665) equal to 1.855921454 and -1.855921454.

Subproblem 1

x + -0.6666666665 = 1.855921454 Simplifying x + -0.6666666665 = 1.855921454 Reorder the terms: -0.6666666665 + x = 1.855921454 Solving -0.6666666665 + x = 1.855921454 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = 1.855921454 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = 1.855921454 + 0.6666666665 x = 1.855921454 + 0.6666666665 Combine like terms: 1.855921454 + 0.6666666665 = 2.5225881205 x = 2.5225881205 Simplifying x = 2.5225881205

Subproblem 2

x + -0.6666666665 = -1.855921454 Simplifying x + -0.6666666665 = -1.855921454 Reorder the terms: -0.6666666665 + x = -1.855921454 Solving -0.6666666665 + x = -1.855921454 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = -1.855921454 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = -1.855921454 + 0.6666666665 x = -1.855921454 + 0.6666666665 Combine like terms: -1.855921454 + 0.6666666665 = -1.1892547875 x = -1.1892547875 Simplifying x = -1.1892547875

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.5225881205, -1.1892547875}

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