9r^2-10r-81=0

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Solution for 9r^2-10r-81=0 equation:


Simplifying
9r2 + -10r + -81 = 0

Reorder the terms:
-81 + -10r + 9r2 = 0

Solving
-81 + -10r + 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'.
-9 + -1.111111111r + r2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + -1.111111111r + 9 + r2 = 0 + 9

Reorder the terms:
-9 + 9 + -1.111111111r + r2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -1.111111111r + r2 = 0 + 9
-1.111111111r + r2 = 0 + 9

Combine like terms: 0 + 9 = 9
-1.111111111r + r2 = 9

The r term is -1.111111111r.  Take half its coefficient (-0.5555555555).
Square it (0.3086419752) and add it to both sides.

Add '0.3086419752' to each side of the equation.
-1.111111111r + 0.3086419752 + r2 = 9 + 0.3086419752

Reorder the terms:
0.3086419752 + -1.111111111r + r2 = 9 + 0.3086419752

Combine like terms: 9 + 0.3086419752 = 9.3086419752
0.3086419752 + -1.111111111r + r2 = 9.3086419752

Factor a perfect square on the left side:
(r + -0.5555555555)(r + -0.5555555555) = 9.3086419752

Calculate the square root of the right side: 3.051006715

Break this problem into two subproblems by setting 
(r + -0.5555555555) equal to 3.051006715 and -3.051006715.

Subproblem 1

r + -0.5555555555 = 3.051006715 Simplifying r + -0.5555555555 = 3.051006715 Reorder the terms: -0.5555555555 + r = 3.051006715 Solving -0.5555555555 + r = 3.051006715 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + r = 3.051006715 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + r = 3.051006715 + 0.5555555555 r = 3.051006715 + 0.5555555555 Combine like terms: 3.051006715 + 0.5555555555 = 3.6065622705 r = 3.6065622705 Simplifying r = 3.6065622705

Subproblem 2

r + -0.5555555555 = -3.051006715 Simplifying r + -0.5555555555 = -3.051006715 Reorder the terms: -0.5555555555 + r = -3.051006715 Solving -0.5555555555 + r = -3.051006715 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + r = -3.051006715 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + r = -3.051006715 + 0.5555555555 r = -3.051006715 + 0.5555555555 Combine like terms: -3.051006715 + 0.5555555555 = -2.4954511595 r = -2.4954511595 Simplifying r = -2.4954511595

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.6065622705, -2.4954511595}

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