0=9x^2-24x+13

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Solution for 0=9x^2-24x+13 equation:


Simplifying
0 = 9x2 + -24x + 13

Reorder the terms:
0 = 13 + -24x + 9x2

Solving
0 = 13 + -24x + 9x2

Solving for variable 'x'.

Combine like terms: 0 + -13 = -13
-13 + 24x + -9x2 = 13 + -24x + 9x2 + -13 + 24x + -9x2

Reorder the terms:
-13 + 24x + -9x2 = 13 + -13 + -24x + 24x + 9x2 + -9x2

Combine like terms: 13 + -13 = 0
-13 + 24x + -9x2 = 0 + -24x + 24x + 9x2 + -9x2
-13 + 24x + -9x2 = -24x + 24x + 9x2 + -9x2

Combine like terms: -24x + 24x = 0
-13 + 24x + -9x2 = 0 + 9x2 + -9x2
-13 + 24x + -9x2 = 9x2 + -9x2

Combine like terms: 9x2 + -9x2 = 0
-13 + 24x + -9x2 = 0

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

Divide each side by '-9'.
1.444444444 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '-1.444444444' to each side of the equation.
1.444444444 + -2.666666667x + -1.444444444 + x2 = 0 + -1.444444444

Reorder the terms:
1.444444444 + -1.444444444 + -2.666666667x + x2 = 0 + -1.444444444

Combine like terms: 1.444444444 + -1.444444444 = 0.000000000
0.000000000 + -2.666666667x + x2 = 0 + -1.444444444
-2.666666667x + x2 = 0 + -1.444444444

Combine like terms: 0 + -1.444444444 = -1.444444444
-2.666666667x + x2 = -1.444444444

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 = -1.444444444 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = -1.444444444 + 1.777777780

Combine like terms: -1.444444444 + 1.777777780 = 0.333333336
1.777777780 + -2.666666667x + x2 = 0.333333336

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 0.333333336

Calculate the square root of the right side: 0.577350271

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 0.577350271 and -0.577350271.

Subproblem 1

x + -1.333333334 = 0.577350271 Simplifying x + -1.333333334 = 0.577350271 Reorder the terms: -1.333333334 + x = 0.577350271 Solving -1.333333334 + x = 0.577350271 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 = 0.577350271 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 0.577350271 + 1.333333334 x = 0.577350271 + 1.333333334 Combine like terms: 0.577350271 + 1.333333334 = 1.910683605 x = 1.910683605 Simplifying x = 1.910683605

Subproblem 2

x + -1.333333334 = -0.577350271 Simplifying x + -1.333333334 = -0.577350271 Reorder the terms: -1.333333334 + x = -0.577350271 Solving -1.333333334 + x = -0.577350271 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 = -0.577350271 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -0.577350271 + 1.333333334 x = -0.577350271 + 1.333333334 Combine like terms: -0.577350271 + 1.333333334 = 0.755983063 x = 0.755983063 Simplifying x = 0.755983063

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.910683605, 0.755983063}

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