4r^2+2r-5=0

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Solution for 4r^2+2r-5=0 equation:


Simplifying
4r2 + 2r + -5 = 0

Reorder the terms:
-5 + 2r + 4r2 = 0

Solving
-5 + 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'.
-1.25 + 0.5r + r2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 0.5r + 1.25 + r2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 0.5r + r2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 0.5r + r2 = 0 + 1.25
0.5r + r2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
0.5r + r2 = 1.25

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 = 1.25 + 0.0625

Reorder the terms:
0.0625 + 0.5r + r2 = 1.25 + 0.0625

Combine like terms: 1.25 + 0.0625 = 1.3125
0.0625 + 0.5r + r2 = 1.3125

Factor a perfect square on the left side:
(r + 0.25)(r + 0.25) = 1.3125

Calculate the square root of the right side: 1.145643924

Break this problem into two subproblems by setting 
(r + 0.25) equal to 1.145643924 and -1.145643924.

Subproblem 1

r + 0.25 = 1.145643924 Simplifying r + 0.25 = 1.145643924 Reorder the terms: 0.25 + r = 1.145643924 Solving 0.25 + r = 1.145643924 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 = 1.145643924 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = 1.145643924 + -0.25 r = 1.145643924 + -0.25 Combine like terms: 1.145643924 + -0.25 = 0.895643924 r = 0.895643924 Simplifying r = 0.895643924

Subproblem 2

r + 0.25 = -1.145643924 Simplifying r + 0.25 = -1.145643924 Reorder the terms: 0.25 + r = -1.145643924 Solving 0.25 + r = -1.145643924 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 = -1.145643924 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = -1.145643924 + -0.25 r = -1.145643924 + -0.25 Combine like terms: -1.145643924 + -0.25 = -1.395643924 r = -1.395643924 Simplifying r = -1.395643924

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.895643924, -1.395643924}

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