5r^2+8r=9

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Solution for 5r^2+8r=9 equation:


Simplifying
5r2 + 8r = 9

Reorder the terms:
8r + 5r2 = 9

Solving
8r + 5r2 = 9

Solving for variable 'r'.

Reorder the terms:
-9 + 8r + 5r2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 8r + 5r2 = 0

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

Divide each side by '5'.
-1.8 + 1.6r + r2 = 0

Move the constant term to the right:

Add '1.8' to each side of the equation.
-1.8 + 1.6r + 1.8 + r2 = 0 + 1.8

Reorder the terms:
-1.8 + 1.8 + 1.6r + r2 = 0 + 1.8

Combine like terms: -1.8 + 1.8 = 0.0
0.0 + 1.6r + r2 = 0 + 1.8
1.6r + r2 = 0 + 1.8

Combine like terms: 0 + 1.8 = 1.8
1.6r + r2 = 1.8

The r term is 1.6r.  Take half its coefficient (0.8).
Square it (0.64) and add it to both sides.

Add '0.64' to each side of the equation.
1.6r + 0.64 + r2 = 1.8 + 0.64

Reorder the terms:
0.64 + 1.6r + r2 = 1.8 + 0.64

Combine like terms: 1.8 + 0.64 = 2.44
0.64 + 1.6r + r2 = 2.44

Factor a perfect square on the left side:
(r + 0.8)(r + 0.8) = 2.44

Calculate the square root of the right side: 1.562049935

Break this problem into two subproblems by setting 
(r + 0.8) equal to 1.562049935 and -1.562049935.

Subproblem 1

r + 0.8 = 1.562049935 Simplifying r + 0.8 = 1.562049935 Reorder the terms: 0.8 + r = 1.562049935 Solving 0.8 + r = 1.562049935 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + r = 1.562049935 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + r = 1.562049935 + -0.8 r = 1.562049935 + -0.8 Combine like terms: 1.562049935 + -0.8 = 0.762049935 r = 0.762049935 Simplifying r = 0.762049935

Subproblem 2

r + 0.8 = -1.562049935 Simplifying r + 0.8 = -1.562049935 Reorder the terms: 0.8 + r = -1.562049935 Solving 0.8 + r = -1.562049935 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + r = -1.562049935 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + r = -1.562049935 + -0.8 r = -1.562049935 + -0.8 Combine like terms: -1.562049935 + -0.8 = -2.362049935 r = -2.362049935 Simplifying r = -2.362049935

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.762049935, -2.362049935}

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