14r^2=5-7r

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Solution for 14r^2=5-7r equation:


Simplifying
14r2 = 5 + -7r

Solving
14r2 = 5 + -7r

Solving for variable 'r'.

Reorder the terms:
-5 + 7r + 14r2 = 5 + -7r + -5 + 7r

Reorder the terms:
-5 + 7r + 14r2 = 5 + -5 + -7r + 7r

Combine like terms: 5 + -5 = 0
-5 + 7r + 14r2 = 0 + -7r + 7r
-5 + 7r + 14r2 = -7r + 7r

Combine like terms: -7r + 7r = 0
-5 + 7r + 14r2 = 0

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

Divide each side by '14'.
-0.3571428571 + 0.5r + r2 = 0

Move the constant term to the right:

Add '0.3571428571' to each side of the equation.
-0.3571428571 + 0.5r + 0.3571428571 + r2 = 0 + 0.3571428571

Reorder the terms:
-0.3571428571 + 0.3571428571 + 0.5r + r2 = 0 + 0.3571428571

Combine like terms: -0.3571428571 + 0.3571428571 = 0.0000000000
0.0000000000 + 0.5r + r2 = 0 + 0.3571428571
0.5r + r2 = 0 + 0.3571428571

Combine like terms: 0 + 0.3571428571 = 0.3571428571
0.5r + r2 = 0.3571428571

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 = 0.3571428571 + 0.0625

Reorder the terms:
0.0625 + 0.5r + r2 = 0.3571428571 + 0.0625

Combine like terms: 0.3571428571 + 0.0625 = 0.4196428571
0.0625 + 0.5r + r2 = 0.4196428571

Factor a perfect square on the left side:
(r + 0.25)(r + 0.25) = 0.4196428571

Calculate the square root of the right side: 0.64779847

Break this problem into two subproblems by setting 
(r + 0.25) equal to 0.64779847 and -0.64779847.

Subproblem 1

r + 0.25 = 0.64779847 Simplifying r + 0.25 = 0.64779847 Reorder the terms: 0.25 + r = 0.64779847 Solving 0.25 + r = 0.64779847 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 = 0.64779847 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = 0.64779847 + -0.25 r = 0.64779847 + -0.25 Combine like terms: 0.64779847 + -0.25 = 0.39779847 r = 0.39779847 Simplifying r = 0.39779847

Subproblem 2

r + 0.25 = -0.64779847 Simplifying r + 0.25 = -0.64779847 Reorder the terms: 0.25 + r = -0.64779847 Solving 0.25 + r = -0.64779847 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 = -0.64779847 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = -0.64779847 + -0.25 r = -0.64779847 + -0.25 Combine like terms: -0.64779847 + -0.25 = -0.89779847 r = -0.89779847 Simplifying r = -0.89779847

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.39779847, -0.89779847}

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