9x^2+24x-7=0

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Solution for 9x^2+24x-7=0 equation: 


Simplifying
9x2 + 24x + -7 = 0

Reorder the terms:
-7 + 24x + 9x2 = 0

Solving
-7 + 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'.
-0.7777777778 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '0.7777777778' to each side of the equation.
-0.7777777778 + 2.666666667x + 0.7777777778 + x2 = 0 + 0.7777777778

Reorder the terms:
-0.7777777778 + 0.7777777778 + 2.666666667x + x2 = 0 + 0.7777777778

Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000
0.0000000000 + 2.666666667x + x2 = 0 + 0.7777777778
2.666666667x + x2 = 0 + 0.7777777778

Combine like terms: 0 + 0.7777777778 = 0.7777777778
2.666666667x + x2 = 0.7777777778

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 = 0.7777777778 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = 0.7777777778 + 1.777777780

Combine like terms: 0.7777777778 + 1.777777780 = 2.5555555578
1.777777780 + 2.666666667x + x2 = 2.5555555578

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 2.5555555578

Calculate the square root of the right side: 1.598610508

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 1.598610508 and -1.598610508.


Subproblem 1
x + 1.333333334 = 1.598610508

Simplifying
x + 1.333333334 = 1.598610508

Reorder the terms:
1.333333334 + x = 1.598610508

Solving
1.333333334 + x = 1.598610508

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 = 1.598610508 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = 1.598610508 + -1.333333334
x = 1.598610508 + -1.333333334

Combine like terms: 1.598610508 + -1.333333334 = 0.265277174
x = 0.265277174

Simplifying
x = 0.265277174


Subproblem 2
x + 1.333333334 = -1.598610508

Simplifying
x + 1.333333334 = -1.598610508

Reorder the terms:
1.333333334 + x = -1.598610508

Solving
1.333333334 + x = -1.598610508

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 = -1.598610508 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = -1.598610508 + -1.333333334
x = -1.598610508 + -1.333333334

Combine like terms: -1.598610508 + -1.333333334 = -2.931943842
x = -2.931943842

Simplifying
x = -2.931943842


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.265277174, -2.931943842}

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