8-(r^2+3r)=0

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Solution for 8-(r^2+3r)=0 equation:


Simplifying
8 + -1(r2 + 3r) = 0

Reorder the terms:
8 + -1(3r + r2) = 0
8 + (3r * -1 + r2 * -1) = 0
8 + (-3r + -1r2) = 0

Solving
8 + -3r + -1r2 = 0

Solving for variable 'r'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-8 + 3r + r2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 3r + 8 + r2 = 0 + 8

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

Combine like terms: -8 + 8 = 0
0 + 3r + r2 = 0 + 8
3r + r2 = 0 + 8

Combine like terms: 0 + 8 = 8
3r + r2 = 8

The r term is 3r.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3r + 2.25 + r2 = 8 + 2.25

Reorder the terms:
2.25 + 3r + r2 = 8 + 2.25

Combine like terms: 8 + 2.25 = 10.25
2.25 + 3r + r2 = 10.25

Factor a perfect square on the left side:
(r + 1.5)(r + 1.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(r + 1.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

r + 1.5 = 3.201562119 Simplifying r + 1.5 = 3.201562119 Reorder the terms: 1.5 + r = 3.201562119 Solving 1.5 + r = 3.201562119 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = 3.201562119 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 3.201562119 + -1.5 r = 3.201562119 + -1.5 Combine like terms: 3.201562119 + -1.5 = 1.701562119 r = 1.701562119 Simplifying r = 1.701562119

Subproblem 2

r + 1.5 = -3.201562119 Simplifying r + 1.5 = -3.201562119 Reorder the terms: 1.5 + r = -3.201562119 Solving 1.5 + r = -3.201562119 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = -3.201562119 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -3.201562119 + -1.5 r = -3.201562119 + -1.5 Combine like terms: -3.201562119 + -1.5 = -4.701562119 r = -4.701562119 Simplifying r = -4.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.701562119, -4.701562119}

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