3x^2-10x-9=0

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Solution for 3x^2-10x-9=0 equation:


Simplifying
3x2 + -10x + -9 = 0

Reorder the terms:
-9 + -10x + 3x2 = 0

Solving
-9 + -10x + 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'.
-3 + -3.333333333x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -3.333333333x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + -3.333333333x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -3.333333333x + x2 = 0 + 3
-3.333333333x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
-3.333333333x + x2 = 3

The x term is -3.333333333x.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333x + 2.777777779 + x2 = 3 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333x + x2 = 3 + 2.777777779

Combine like terms: 3 + 2.777777779 = 5.777777779
2.777777779 + -3.333333333x + x2 = 5.777777779

Factor a perfect square on the left side:
(x + -1.666666667)(x + -1.666666667) = 5.777777779

Calculate the square root of the right side: 2.403700851

Break this problem into two subproblems by setting 
(x + -1.666666667) equal to 2.403700851 and -2.403700851.

Subproblem 1

x + -1.666666667 = 2.403700851 Simplifying x + -1.666666667 = 2.403700851 Reorder the terms: -1.666666667 + x = 2.403700851 Solving -1.666666667 + x = 2.403700851 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = 2.403700851 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = 2.403700851 + 1.666666667 x = 2.403700851 + 1.666666667 Combine like terms: 2.403700851 + 1.666666667 = 4.070367518 x = 4.070367518 Simplifying x = 4.070367518

Subproblem 2

x + -1.666666667 = -2.403700851 Simplifying x + -1.666666667 = -2.403700851 Reorder the terms: -1.666666667 + x = -2.403700851 Solving -1.666666667 + x = -2.403700851 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = -2.403700851 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = -2.403700851 + 1.666666667 x = -2.403700851 + 1.666666667 Combine like terms: -2.403700851 + 1.666666667 = -0.737034184 x = -0.737034184 Simplifying x = -0.737034184

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.070367518, -0.737034184}

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