3r^2-8r-9=0

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Solution for 3r^2-8r-9=0 equation:


Simplifying
3r2 + -8r + -9 = 0

Reorder the terms:
-9 + -8r + 3r2 = 0

Solving
-9 + -8r + 3r2 = 0

Solving for variable 'r'.

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

Divide each side by '3'.
-3 + -2.666666667r + r2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -2.666666667r + 3 + r2 = 0 + 3

Reorder the terms:
-3 + 3 + -2.666666667r + r2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -2.666666667r + r2 = 0 + 3
-2.666666667r + r2 = 0 + 3

Combine like terms: 0 + 3 = 3
-2.666666667r + r2 = 3

The r term is -2.666666667r.  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.
-2.666666667r + 1.777777780 + r2 = 3 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667r + r2 = 3 + 1.777777780

Combine like terms: 3 + 1.777777780 = 4.77777778
1.777777780 + -2.666666667r + r2 = 4.77777778

Factor a perfect square on the left side:
(r + -1.333333334)(r + -1.333333334) = 4.77777778

Calculate the square root of the right side: 2.185812842

Break this problem into two subproblems by setting 
(r + -1.333333334) equal to 2.185812842 and -2.185812842.

Subproblem 1

r + -1.333333334 = 2.185812842 Simplifying r + -1.333333334 = 2.185812842 Reorder the terms: -1.333333334 + r = 2.185812842 Solving -1.333333334 + r = 2.185812842 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + r = 2.185812842 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + r = 2.185812842 + 1.333333334 r = 2.185812842 + 1.333333334 Combine like terms: 2.185812842 + 1.333333334 = 3.519146176 r = 3.519146176 Simplifying r = 3.519146176

Subproblem 2

r + -1.333333334 = -2.185812842 Simplifying r + -1.333333334 = -2.185812842 Reorder the terms: -1.333333334 + r = -2.185812842 Solving -1.333333334 + r = -2.185812842 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + r = -2.185812842 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + r = -2.185812842 + 1.333333334 r = -2.185812842 + 1.333333334 Combine like terms: -2.185812842 + 1.333333334 = -0.852479508 r = -0.852479508 Simplifying r = -0.852479508

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.519146176, -0.852479508}

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