5r^2-r-2=0

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


Simplifying
5r2 + -1r + -2 = 0

Reorder the terms:
-2 + -1r + 5r2 = 0

Solving
-2 + -1r + 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'.
-0.4 + -0.2r + r2 = 0

Move the constant term to the right:

Add '0.4' to each side of the equation.
-0.4 + -0.2r + 0.4 + r2 = 0 + 0.4

Reorder the terms:
-0.4 + 0.4 + -0.2r + r2 = 0 + 0.4

Combine like terms: -0.4 + 0.4 = 0.0
0.0 + -0.2r + r2 = 0 + 0.4
-0.2r + r2 = 0 + 0.4

Combine like terms: 0 + 0.4 = 0.4
-0.2r + r2 = 0.4

The r term is -0.2r.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2r + 0.01 + r2 = 0.4 + 0.01

Reorder the terms:
0.01 + -0.2r + r2 = 0.4 + 0.01

Combine like terms: 0.4 + 0.01 = 0.41
0.01 + -0.2r + r2 = 0.41

Factor a perfect square on the left side:
(r + -0.1)(r + -0.1) = 0.41

Calculate the square root of the right side: 0.640312424

Break this problem into two subproblems by setting 
(r + -0.1) equal to 0.640312424 and -0.640312424.

Subproblem 1

r + -0.1 = 0.640312424 Simplifying r + -0.1 = 0.640312424 Reorder the terms: -0.1 + r = 0.640312424 Solving -0.1 + r = 0.640312424 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + r = 0.640312424 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + r = 0.640312424 + 0.1 r = 0.640312424 + 0.1 Combine like terms: 0.640312424 + 0.1 = 0.740312424 r = 0.740312424 Simplifying r = 0.740312424

Subproblem 2

r + -0.1 = -0.640312424 Simplifying r + -0.1 = -0.640312424 Reorder the terms: -0.1 + r = -0.640312424 Solving -0.1 + r = -0.640312424 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + r = -0.640312424 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + r = -0.640312424 + 0.1 r = -0.640312424 + 0.1 Combine like terms: -0.640312424 + 0.1 = -0.540312424 r = -0.540312424 Simplifying r = -0.540312424

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.740312424, -0.540312424}

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