2r^2+1=5r

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Solution for 2r^2+1=5r equation:


Simplifying
2r2 + 1 = 5r

Reorder the terms:
1 + 2r2 = 5r

Solving
1 + 2r2 = 5r

Solving for variable 'r'.

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

Combine like terms: 5r + -5r = 0
1 + -5r + 2r2 = 0

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

Divide each side by '2'.
0.5 + -2.5r + r2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2.5r + -0.5 + r2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2.5r + r2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2.5r + r2 = 0 + -0.5
-2.5r + r2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2.5r + r2 = -0.5

The r term is -2.5r.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5r + 1.5625 + r2 = -0.5 + 1.5625

Reorder the terms:
1.5625 + -2.5r + r2 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + -2.5r + r2 = 1.0625

Factor a perfect square on the left side:
(r + -1.25)(r + -1.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(r + -1.25) equal to 1.030776406 and -1.030776406.

Subproblem 1

r + -1.25 = 1.030776406 Simplifying r + -1.25 = 1.030776406 Reorder the terms: -1.25 + r = 1.030776406 Solving -1.25 + r = 1.030776406 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + r = 1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + r = 1.030776406 + 1.25 r = 1.030776406 + 1.25 Combine like terms: 1.030776406 + 1.25 = 2.280776406 r = 2.280776406 Simplifying r = 2.280776406

Subproblem 2

r + -1.25 = -1.030776406 Simplifying r + -1.25 = -1.030776406 Reorder the terms: -1.25 + r = -1.030776406 Solving -1.25 + r = -1.030776406 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + r = -1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + r = -1.030776406 + 1.25 r = -1.030776406 + 1.25 Combine like terms: -1.030776406 + 1.25 = 0.219223594 r = 0.219223594 Simplifying r = 0.219223594

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.280776406, 0.219223594}

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