0=3x^2-7x-11

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Solution for 0=3x^2-7x-11 equation:


Simplifying
0 = 3x2 + -7x + -11

Reorder the terms:
0 = -11 + -7x + 3x2

Solving
0 = -11 + -7x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + 11 = 11
11 + 7x + -3x2 = -11 + -7x + 3x2 + 11 + 7x + -3x2

Reorder the terms:
11 + 7x + -3x2 = -11 + 11 + -7x + 7x + 3x2 + -3x2

Combine like terms: -11 + 11 = 0
11 + 7x + -3x2 = 0 + -7x + 7x + 3x2 + -3x2
11 + 7x + -3x2 = -7x + 7x + 3x2 + -3x2

Combine like terms: -7x + 7x = 0
11 + 7x + -3x2 = 0 + 3x2 + -3x2
11 + 7x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
11 + 7x + -3x2 = 0

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

Divide each side by '-3'.
-3.666666667 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + -2.333333333x + 3.666666667 + x2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + -2.333333333x + x2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + -2.333333333x + x2 = 0 + 3.666666667
-2.333333333x + x2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
-2.333333333x + x2 = 3.666666667

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = 3.666666667 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 3.666666667 + 1.361111112

Combine like terms: 3.666666667 + 1.361111112 = 5.027777779
1.361111112 + -2.333333333x + x2 = 5.027777779

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 5.027777779

Calculate the square root of the right side: 2.242270675

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 2.242270675 and -2.242270675.

Subproblem 1

x + -1.166666667 = 2.242270675 Simplifying x + -1.166666667 = 2.242270675 Reorder the terms: -1.166666667 + x = 2.242270675 Solving -1.166666667 + x = 2.242270675 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = 2.242270675 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = 2.242270675 + 1.166666667 x = 2.242270675 + 1.166666667 Combine like terms: 2.242270675 + 1.166666667 = 3.408937342 x = 3.408937342 Simplifying x = 3.408937342

Subproblem 2

x + -1.166666667 = -2.242270675 Simplifying x + -1.166666667 = -2.242270675 Reorder the terms: -1.166666667 + x = -2.242270675 Solving -1.166666667 + x = -2.242270675 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = -2.242270675 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = -2.242270675 + 1.166666667 x = -2.242270675 + 1.166666667 Combine like terms: -2.242270675 + 1.166666667 = -1.075604008 x = -1.075604008 Simplifying x = -1.075604008

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.408937342, -1.075604008}

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