3x^2-11x-3=0

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


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

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

Solving
-3 + -11x + 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'.
-1 + -3.666666667x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3.666666667x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -3.666666667x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3.666666667x + x2 = 0 + 1
-3.666666667x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3.666666667x + x2 = 1

The x term is -3.666666667x.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667x + 3.361111114 + x2 = 1 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667x + x2 = 1 + 3.361111114

Combine like terms: 1 + 3.361111114 = 4.361111114
3.361111114 + -3.666666667x + x2 = 4.361111114

Factor a perfect square on the left side:
(x + -1.833333334)(x + -1.833333334) = 4.361111114

Calculate the square root of the right side: 2.088327348

Break this problem into two subproblems by setting 
(x + -1.833333334) equal to 2.088327348 and -2.088327348.

Subproblem 1

x + -1.833333334 = 2.088327348 Simplifying x + -1.833333334 = 2.088327348 Reorder the terms: -1.833333334 + x = 2.088327348 Solving -1.833333334 + x = 2.088327348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + x = 2.088327348 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = 2.088327348 + 1.833333334 x = 2.088327348 + 1.833333334 Combine like terms: 2.088327348 + 1.833333334 = 3.921660682 x = 3.921660682 Simplifying x = 3.921660682

Subproblem 2

x + -1.833333334 = -2.088327348 Simplifying x + -1.833333334 = -2.088327348 Reorder the terms: -1.833333334 + x = -2.088327348 Solving -1.833333334 + x = -2.088327348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + x = -2.088327348 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = -2.088327348 + 1.833333334 x = -2.088327348 + 1.833333334 Combine like terms: -2.088327348 + 1.833333334 = -0.254994014 x = -0.254994014 Simplifying x = -0.254994014

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.921660682, -0.254994014}

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