5r^2+5r-1=0

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Solution for 5r^2+5r-1=0 equation:


Simplifying
5r2 + 5r + -1 = 0

Reorder the terms:
-1 + 5r + 5r2 = 0

Solving
-1 + 5r + 5r2 = 0

Solving for variable 'r'.

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

Divide each side by '5'.
-0.2 + r + r2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + r + 0.2 + r2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + r + r2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + r + r2 = 0 + 0.2
r + r2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
r + r2 = 0.2

The r term is r.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + r2 = 0.2 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + r2 = 0.2 + 0.25

Combine like terms: 0.2 + 0.25 = 0.45
1.25 + r2 = 0.45

Factor a perfect square on the left side:
(r + 0.5)(r + 0.5) = 0.45

Calculate the square root of the right side: 0.670820393

Break this problem into two subproblems by setting 
(r + 0.5) equal to 0.670820393 and -0.670820393.

Subproblem 1

r + 0.5 = 0.670820393 Simplifying r + 0.5 = 0.670820393 Reorder the terms: 0.5 + r = 0.670820393 Solving 0.5 + r = 0.670820393 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 0.670820393 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.670820393 + -0.5 r = 0.670820393 + -0.5 Combine like terms: 0.670820393 + -0.5 = 0.170820393 r = 0.170820393 Simplifying r = 0.170820393

Subproblem 2

r + 0.5 = -0.670820393 Simplifying r + 0.5 = -0.670820393 Reorder the terms: 0.5 + r = -0.670820393 Solving 0.5 + r = -0.670820393 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -0.670820393 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.670820393 + -0.5 r = -0.670820393 + -0.5 Combine like terms: -0.670820393 + -0.5 = -1.170820393 r = -1.170820393 Simplifying r = -1.170820393

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.170820393, -1.170820393}

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