0=3r^2+5r-1

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Solution for 0=3r^2+5r-1 equation:


Simplifying
0 = 3r2 + 5r + -1

Reorder the terms:
0 = -1 + 5r + 3r2

Solving
0 = -1 + 5r + 3r2

Solving for variable 'r'.

Combine like terms: 0 + 1 = 1
1 + -5r + -3r2 = -1 + 5r + 3r2 + 1 + -5r + -3r2

Reorder the terms:
1 + -5r + -3r2 = -1 + 1 + 5r + -5r + 3r2 + -3r2

Combine like terms: -1 + 1 = 0
1 + -5r + -3r2 = 0 + 5r + -5r + 3r2 + -3r2
1 + -5r + -3r2 = 5r + -5r + 3r2 + -3r2

Combine like terms: 5r + -5r = 0
1 + -5r + -3r2 = 0 + 3r2 + -3r2
1 + -5r + -3r2 = 3r2 + -3r2

Combine like terms: 3r2 + -3r2 = 0
1 + -5r + -3r2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-0.3333333333 + 1.666666667r + r2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 1.666666667r + 0.3333333333 + r2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 1.666666667r + r2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 1.666666667r + r2 = 0 + 0.3333333333
1.666666667r + r2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
1.666666667r + r2 = 0.3333333333

The r term is 1.666666667r.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667r + 0.6944444447 + r2 = 0.3333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667r + r2 = 0.3333333333 + 0.6944444447

Combine like terms: 0.3333333333 + 0.6944444447 = 1.027777778
0.6944444447 + 1.666666667r + r2 = 1.027777778

Factor a perfect square on the left side:
(r + 0.8333333335)(r + 0.8333333335) = 1.027777778

Calculate the square root of the right side: 1.013793755

Break this problem into two subproblems by setting 
(r + 0.8333333335) equal to 1.013793755 and -1.013793755.

Subproblem 1

r + 0.8333333335 = 1.013793755 Simplifying r + 0.8333333335 = 1.013793755 Reorder the terms: 0.8333333335 + r = 1.013793755 Solving 0.8333333335 + r = 1.013793755 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + r = 1.013793755 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + r = 1.013793755 + -0.8333333335 r = 1.013793755 + -0.8333333335 Combine like terms: 1.013793755 + -0.8333333335 = 0.1804604215 r = 0.1804604215 Simplifying r = 0.1804604215

Subproblem 2

r + 0.8333333335 = -1.013793755 Simplifying r + 0.8333333335 = -1.013793755 Reorder the terms: 0.8333333335 + r = -1.013793755 Solving 0.8333333335 + r = -1.013793755 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + r = -1.013793755 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + r = -1.013793755 + -0.8333333335 r = -1.013793755 + -0.8333333335 Combine like terms: -1.013793755 + -0.8333333335 = -1.8471270885 r = -1.8471270885 Simplifying r = -1.8471270885

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.1804604215, -1.8471270885}

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