9r^2-6r-7=0

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Solution for 9r^2-6r-7=0 equation: 


Simplifying
9r2 + -6r + -7 = 0

Reorder the terms:
-7 + -6r + 9r2 = 0

Solving
-7 + -6r + 9r2 = 0

Solving for variable 'r'.

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

Divide each side by '9'.
-0.7777777778 + -0.6666666667r + r2 = 0

Move the constant term to the right:

Add '0.7777777778' to each side of the equation.
-0.7777777778 + -0.6666666667r + 0.7777777778 + r2 = 0 + 0.7777777778

Reorder the terms:
-0.7777777778 + 0.7777777778 + -0.6666666667r + r2 = 0 + 0.7777777778

Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000
0.0000000000 + -0.6666666667r + r2 = 0 + 0.7777777778
-0.6666666667r + r2 = 0 + 0.7777777778

Combine like terms: 0 + 0.7777777778 = 0.7777777778
-0.6666666667r + r2 = 0.7777777778

The r term is -0.6666666667r.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667r + 0.1111111112 + r2 = 0.7777777778 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667r + r2 = 0.7777777778 + 0.1111111112

Combine like terms: 0.7777777778 + 0.1111111112 = 0.888888889
0.1111111112 + -0.6666666667r + r2 = 0.888888889

Factor a perfect square on the left side:
(r + -0.3333333334)(r + -0.3333333334) = 0.888888889

Calculate the square root of the right side: 0.942809042

Break this problem into two subproblems by setting 
(r + -0.3333333334) equal to 0.942809042 and -0.942809042.


Subproblem 1
r + -0.3333333334 = 0.942809042

Simplifying
r + -0.3333333334 = 0.942809042

Reorder the terms:
-0.3333333334 + r = 0.942809042

Solving
-0.3333333334 + r = 0.942809042

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + r = 0.942809042 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + r = 0.942809042 + 0.3333333334
r = 0.942809042 + 0.3333333334

Combine like terms: 0.942809042 + 0.3333333334 = 1.2761423754
r = 1.2761423754

Simplifying
r = 1.2761423754


Subproblem 2
r + -0.3333333334 = -0.942809042

Simplifying
r + -0.3333333334 = -0.942809042

Reorder the terms:
-0.3333333334 + r = -0.942809042

Solving
-0.3333333334 + r = -0.942809042

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + r = -0.942809042 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + r = -0.942809042 + 0.3333333334
r = -0.942809042 + 0.3333333334

Combine like terms: -0.942809042 + 0.3333333334 = -0.6094757086
r = -0.6094757086

Simplifying
r = -0.6094757086


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.2761423754, -0.6094757086}

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