8r^2=19-12r

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Solution for 8r^2=19-12r equation:


Simplifying
8r2 = 19 + -12r

Solving
8r2 = 19 + -12r

Solving for variable 'r'.

Reorder the terms:
-19 + 12r + 8r2 = 19 + -12r + -19 + 12r

Reorder the terms:
-19 + 12r + 8r2 = 19 + -19 + -12r + 12r

Combine like terms: 19 + -19 = 0
-19 + 12r + 8r2 = 0 + -12r + 12r
-19 + 12r + 8r2 = -12r + 12r

Combine like terms: -12r + 12r = 0
-19 + 12r + 8r2 = 0

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

Divide each side by '8'.
-2.375 + 1.5r + r2 = 0

Move the constant term to the right:

Add '2.375' to each side of the equation.
-2.375 + 1.5r + 2.375 + r2 = 0 + 2.375

Reorder the terms:
-2.375 + 2.375 + 1.5r + r2 = 0 + 2.375

Combine like terms: -2.375 + 2.375 = 0.000
0.000 + 1.5r + r2 = 0 + 2.375
1.5r + r2 = 0 + 2.375

Combine like terms: 0 + 2.375 = 2.375
1.5r + r2 = 2.375

The r term is 1.5r.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5r + 0.5625 + r2 = 2.375 + 0.5625

Reorder the terms:
0.5625 + 1.5r + r2 = 2.375 + 0.5625

Combine like terms: 2.375 + 0.5625 = 2.9375
0.5625 + 1.5r + r2 = 2.9375

Factor a perfect square on the left side:
(r + 0.75)(r + 0.75) = 2.9375

Calculate the square root of the right side: 1.71391365

Break this problem into two subproblems by setting 
(r + 0.75) equal to 1.71391365 and -1.71391365.

Subproblem 1

r + 0.75 = 1.71391365 Simplifying r + 0.75 = 1.71391365 Reorder the terms: 0.75 + r = 1.71391365 Solving 0.75 + r = 1.71391365 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = 1.71391365 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = 1.71391365 + -0.75 r = 1.71391365 + -0.75 Combine like terms: 1.71391365 + -0.75 = 0.96391365 r = 0.96391365 Simplifying r = 0.96391365

Subproblem 2

r + 0.75 = -1.71391365 Simplifying r + 0.75 = -1.71391365 Reorder the terms: 0.75 + r = -1.71391365 Solving 0.75 + r = -1.71391365 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = -1.71391365 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = -1.71391365 + -0.75 r = -1.71391365 + -0.75 Combine like terms: -1.71391365 + -0.75 = -2.46391365 r = -2.46391365 Simplifying r = -2.46391365

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.96391365, -2.46391365}

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