8r^2=r+6

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Solution for 8r^2=r+6 equation:


Simplifying
8r2 = r + 6

Reorder the terms:
8r2 = 6 + r

Solving
8r2 = 6 + r

Solving for variable 'r'.

Reorder the terms:
-6 + -1r + 8r2 = 6 + r + -6 + -1r

Reorder the terms:
-6 + -1r + 8r2 = 6 + -6 + r + -1r

Combine like terms: 6 + -6 = 0
-6 + -1r + 8r2 = 0 + r + -1r
-6 + -1r + 8r2 = r + -1r

Combine like terms: r + -1r = 0
-6 + -1r + 8r2 = 0

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

Divide each side by '8'.
-0.75 + -0.125r + r2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + -0.125r + 0.75 + r2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + -0.125r + r2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + -0.125r + r2 = 0 + 0.75
-0.125r + r2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
-0.125r + r2 = 0.75

The r term is -0.125r.  Take half its coefficient (-0.0625).
Square it (0.00390625) and add it to both sides.

Add '0.00390625' to each side of the equation.
-0.125r + 0.00390625 + r2 = 0.75 + 0.00390625

Reorder the terms:
0.00390625 + -0.125r + r2 = 0.75 + 0.00390625

Combine like terms: 0.75 + 0.00390625 = 0.75390625
0.00390625 + -0.125r + r2 = 0.75390625

Factor a perfect square on the left side:
(r + -0.0625)(r + -0.0625) = 0.75390625

Calculate the square root of the right side: 0.868277749

Break this problem into two subproblems by setting 
(r + -0.0625) equal to 0.868277749 and -0.868277749.

Subproblem 1

r + -0.0625 = 0.868277749 Simplifying r + -0.0625 = 0.868277749 Reorder the terms: -0.0625 + r = 0.868277749 Solving -0.0625 + r = 0.868277749 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + r = 0.868277749 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + r = 0.868277749 + 0.0625 r = 0.868277749 + 0.0625 Combine like terms: 0.868277749 + 0.0625 = 0.930777749 r = 0.930777749 Simplifying r = 0.930777749

Subproblem 2

r + -0.0625 = -0.868277749 Simplifying r + -0.0625 = -0.868277749 Reorder the terms: -0.0625 + r = -0.868277749 Solving -0.0625 + r = -0.868277749 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + r = -0.868277749 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + r = -0.868277749 + 0.0625 r = -0.868277749 + 0.0625 Combine like terms: -0.868277749 + 0.0625 = -0.805777749 r = -0.805777749 Simplifying r = -0.805777749

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.930777749, -0.805777749}

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