4r^2+2r-1=0

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Solution for 4r^2+2r-1=0 equation:


Simplifying
4r2 + 2r + -1 = 0

Reorder the terms:
-1 + 2r + 4r2 = 0

Solving
-1 + 2r + 4r2 = 0

Solving for variable 'r'.

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

Divide each side by '4'.
-0.25 + 0.5r + r2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 0.5r + 0.25 + r2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 0.5r + r2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 0.5r + r2 = 0 + 0.25
0.5r + r2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
0.5r + r2 = 0.25

The r term is 0.5r.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5r + 0.0625 + r2 = 0.25 + 0.0625

Reorder the terms:
0.0625 + 0.5r + r2 = 0.25 + 0.0625

Combine like terms: 0.25 + 0.0625 = 0.3125
0.0625 + 0.5r + r2 = 0.3125

Factor a perfect square on the left side:
(r + 0.25)(r + 0.25) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(r + 0.25) equal to 0.559016994 and -0.559016994.

Subproblem 1

r + 0.25 = 0.559016994 Simplifying r + 0.25 = 0.559016994 Reorder the terms: 0.25 + r = 0.559016994 Solving 0.25 + r = 0.559016994 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + r = 0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = 0.559016994 + -0.25 r = 0.559016994 + -0.25 Combine like terms: 0.559016994 + -0.25 = 0.309016994 r = 0.309016994 Simplifying r = 0.309016994

Subproblem 2

r + 0.25 = -0.559016994 Simplifying r + 0.25 = -0.559016994 Reorder the terms: 0.25 + r = -0.559016994 Solving 0.25 + r = -0.559016994 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + r = -0.559016994 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = -0.559016994 + -0.25 r = -0.559016994 + -0.25 Combine like terms: -0.559016994 + -0.25 = -0.809016994 r = -0.809016994 Simplifying r = -0.809016994

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.309016994, -0.809016994}

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