2r^2-r-5.25=0

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Solution for 2r^2-r-5.25=0 equation:


Simplifying
2r2 + -1r + -5.25 = 0

Reorder the terms:
-5.25 + -1r + 2r2 = 0

Solving
-5.25 + -1r + 2r2 = 0

Solving for variable 'r'.

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

Divide each side by '2'.
-2.625 + -0.5r + r2 = 0

Move the constant term to the right:

Add '2.625' to each side of the equation.
-2.625 + -0.5r + 2.625 + r2 = 0 + 2.625

Reorder the terms:
-2.625 + 2.625 + -0.5r + r2 = 0 + 2.625

Combine like terms: -2.625 + 2.625 = 0.000
0.000 + -0.5r + r2 = 0 + 2.625
-0.5r + r2 = 0 + 2.625

Combine like terms: 0 + 2.625 = 2.625
-0.5r + r2 = 2.625

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 = 2.625 + 0.0625

Reorder the terms:
0.0625 + -0.5r + r2 = 2.625 + 0.0625

Combine like terms: 2.625 + 0.0625 = 2.6875
0.0625 + -0.5r + r2 = 2.6875

Factor a perfect square on the left side:
(r + -0.25)(r + -0.25) = 2.6875

Calculate the square root of the right side: 1.639359631

Break this problem into two subproblems by setting 
(r + -0.25) equal to 1.639359631 and -1.639359631.

Subproblem 1

r + -0.25 = 1.639359631 Simplifying r + -0.25 = 1.639359631 Reorder the terms: -0.25 + r = 1.639359631 Solving -0.25 + r = 1.639359631 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 = 1.639359631 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + r = 1.639359631 + 0.25 r = 1.639359631 + 0.25 Combine like terms: 1.639359631 + 0.25 = 1.889359631 r = 1.889359631 Simplifying r = 1.889359631

Subproblem 2

r + -0.25 = -1.639359631 Simplifying r + -0.25 = -1.639359631 Reorder the terms: -0.25 + r = -1.639359631 Solving -0.25 + r = -1.639359631 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 = -1.639359631 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + r = -1.639359631 + 0.25 r = -1.639359631 + 0.25 Combine like terms: -1.639359631 + 0.25 = -1.389359631 r = -1.389359631 Simplifying r = -1.389359631

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.889359631, -1.389359631}

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