9x^2-24x-2=0

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Solution for 9x^2-24x-2=0 equation:


Simplifying
9x2 + -24x + -2 = 0

Reorder the terms:
-2 + -24x + 9x2 = 0

Solving
-2 + -24x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-0.2222222222 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '0.2222222222' to each side of the equation.
-0.2222222222 + -2.666666667x + 0.2222222222 + x2 = 0 + 0.2222222222

Reorder the terms:
-0.2222222222 + 0.2222222222 + -2.666666667x + x2 = 0 + 0.2222222222

Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000
0.0000000000 + -2.666666667x + x2 = 0 + 0.2222222222
-2.666666667x + x2 = 0 + 0.2222222222

Combine like terms: 0 + 0.2222222222 = 0.2222222222
-2.666666667x + x2 = 0.2222222222

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = 0.2222222222 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = 0.2222222222 + 1.777777780

Combine like terms: 0.2222222222 + 1.777777780 = 2.0000000022
1.777777780 + -2.666666667x + x2 = 2.0000000022

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 2.0000000022

Calculate the square root of the right side: 1.414213563

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 1.414213563 and -1.414213563.

Subproblem 1

x + -1.333333334 = 1.414213563 Simplifying x + -1.333333334 = 1.414213563 Reorder the terms: -1.333333334 + x = 1.414213563 Solving -1.333333334 + x = 1.414213563 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 1.414213563 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 1.414213563 + 1.333333334 x = 1.414213563 + 1.333333334 Combine like terms: 1.414213563 + 1.333333334 = 2.747546897 x = 2.747546897 Simplifying x = 2.747546897

Subproblem 2

x + -1.333333334 = -1.414213563 Simplifying x + -1.333333334 = -1.414213563 Reorder the terms: -1.333333334 + x = -1.414213563 Solving -1.333333334 + x = -1.414213563 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -1.414213563 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -1.414213563 + 1.333333334 x = -1.414213563 + 1.333333334 Combine like terms: -1.414213563 + 1.333333334 = -0.080880229 x = -0.080880229 Simplifying x = -0.080880229

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.747546897, -0.080880229}

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