r^2-9r-38=0

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Solution for r^2-9r-38=0 equation:


Simplifying
r2 + -9r + -38 = 0

Reorder the terms:
-38 + -9r + r2 = 0

Solving
-38 + -9r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '38' to each side of the equation.
-38 + -9r + 38 + r2 = 0 + 38

Reorder the terms:
-38 + 38 + -9r + r2 = 0 + 38

Combine like terms: -38 + 38 = 0
0 + -9r + r2 = 0 + 38
-9r + r2 = 0 + 38

Combine like terms: 0 + 38 = 38
-9r + r2 = 38

The r term is -9r.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9r + 20.25 + r2 = 38 + 20.25

Reorder the terms:
20.25 + -9r + r2 = 38 + 20.25

Combine like terms: 38 + 20.25 = 58.25
20.25 + -9r + r2 = 58.25

Factor a perfect square on the left side:
(r + -4.5)(r + -4.5) = 58.25

Calculate the square root of the right side: 7.632168761

Break this problem into two subproblems by setting 
(r + -4.5) equal to 7.632168761 and -7.632168761.

Subproblem 1

r + -4.5 = 7.632168761 Simplifying r + -4.5 = 7.632168761 Reorder the terms: -4.5 + r = 7.632168761 Solving -4.5 + r = 7.632168761 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + r = 7.632168761 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + r = 7.632168761 + 4.5 r = 7.632168761 + 4.5 Combine like terms: 7.632168761 + 4.5 = 12.132168761 r = 12.132168761 Simplifying r = 12.132168761

Subproblem 2

r + -4.5 = -7.632168761 Simplifying r + -4.5 = -7.632168761 Reorder the terms: -4.5 + r = -7.632168761 Solving -4.5 + r = -7.632168761 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + r = -7.632168761 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + r = -7.632168761 + 4.5 r = -7.632168761 + 4.5 Combine like terms: -7.632168761 + 4.5 = -3.132168761 r = -3.132168761 Simplifying r = -3.132168761

Solution

The solution to the problem is based on the solutions from the subproblems. r = {12.132168761, -3.132168761}

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