9x^2-24x-1=0

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Solution for 9x^2-24x-1=0 equation:


Simplifying
9x2 + -24x + -1 = 0

Reorder the terms:
-1 + -24x + 9x2 = 0

Solving
-1 + -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'.
-0.1111111111 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -2.666666667x + 0.1111111111 + x2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -2.666666667x + x2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -2.666666667x + x2 = 0 + 0.1111111111
-2.666666667x + x2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-2.666666667x + x2 = 0.1111111111

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 = 0.1111111111 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = 0.1111111111 + 1.777777780

Combine like terms: 0.1111111111 + 1.777777780 = 1.8888888911
1.777777780 + -2.666666667x + x2 = 1.8888888911

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 1.8888888911

Calculate the square root of the right side: 1.374368543

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 1.374368543 and -1.374368543.

Subproblem 1

x + -1.333333334 = 1.374368543 Simplifying x + -1.333333334 = 1.374368543 Reorder the terms: -1.333333334 + x = 1.374368543 Solving -1.333333334 + x = 1.374368543 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 = 1.374368543 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 1.374368543 + 1.333333334 x = 1.374368543 + 1.333333334 Combine like terms: 1.374368543 + 1.333333334 = 2.707701877 x = 2.707701877 Simplifying x = 2.707701877

Subproblem 2

x + -1.333333334 = -1.374368543 Simplifying x + -1.333333334 = -1.374368543 Reorder the terms: -1.333333334 + x = -1.374368543 Solving -1.333333334 + x = -1.374368543 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 = -1.374368543 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -1.374368543 + 1.333333334 x = -1.374368543 + 1.333333334 Combine like terms: -1.374368543 + 1.333333334 = -0.041035209 x = -0.041035209 Simplifying x = -0.041035209

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.707701877, -0.041035209}

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