0=16r^2+3r-9

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Solution for 0=16r^2+3r-9 equation:


Simplifying
0 = 16r2 + 3r + -9

Reorder the terms:
0 = -9 + 3r + 16r2

Solving
0 = -9 + 3r + 16r2

Solving for variable 'r'.

Combine like terms: 0 + 9 = 9
9 + -3r + -16r2 = -9 + 3r + 16r2 + 9 + -3r + -16r2

Reorder the terms:
9 + -3r + -16r2 = -9 + 9 + 3r + -3r + 16r2 + -16r2

Combine like terms: -9 + 9 = 0
9 + -3r + -16r2 = 0 + 3r + -3r + 16r2 + -16r2
9 + -3r + -16r2 = 3r + -3r + 16r2 + -16r2

Combine like terms: 3r + -3r = 0
9 + -3r + -16r2 = 0 + 16r2 + -16r2
9 + -3r + -16r2 = 16r2 + -16r2

Combine like terms: 16r2 + -16r2 = 0
9 + -3r + -16r2 = 0

Begin completing the square.  Divide all terms by
-16 the coefficient of the squared term: 

Divide each side by '-16'.
-0.5625 + 0.1875r + r2 = 0

Move the constant term to the right:

Add '0.5625' to each side of the equation.
-0.5625 + 0.1875r + 0.5625 + r2 = 0 + 0.5625

Reorder the terms:
-0.5625 + 0.5625 + 0.1875r + r2 = 0 + 0.5625

Combine like terms: -0.5625 + 0.5625 = 0.0000
0.0000 + 0.1875r + r2 = 0 + 0.5625
0.1875r + r2 = 0 + 0.5625

Combine like terms: 0 + 0.5625 = 0.5625
0.1875r + r2 = 0.5625

The r term is 0.1875r.  Take half its coefficient (0.09375).
Square it (0.0087890625) and add it to both sides.

Add '0.0087890625' to each side of the equation.
0.1875r + 0.0087890625 + r2 = 0.5625 + 0.0087890625

Reorder the terms:
0.0087890625 + 0.1875r + r2 = 0.5625 + 0.0087890625

Combine like terms: 0.5625 + 0.0087890625 = 0.5712890625
0.0087890625 + 0.1875r + r2 = 0.5712890625

Factor a perfect square on the left side:
(r + 0.09375)(r + 0.09375) = 0.5712890625

Calculate the square root of the right side: 0.755836664

Break this problem into two subproblems by setting 
(r + 0.09375) equal to 0.755836664 and -0.755836664.

Subproblem 1

r + 0.09375 = 0.755836664 Simplifying r + 0.09375 = 0.755836664 Reorder the terms: 0.09375 + r = 0.755836664 Solving 0.09375 + r = 0.755836664 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.09375' to each side of the equation. 0.09375 + -0.09375 + r = 0.755836664 + -0.09375 Combine like terms: 0.09375 + -0.09375 = 0.00000 0.00000 + r = 0.755836664 + -0.09375 r = 0.755836664 + -0.09375 Combine like terms: 0.755836664 + -0.09375 = 0.662086664 r = 0.662086664 Simplifying r = 0.662086664

Subproblem 2

r + 0.09375 = -0.755836664 Simplifying r + 0.09375 = -0.755836664 Reorder the terms: 0.09375 + r = -0.755836664 Solving 0.09375 + r = -0.755836664 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.09375' to each side of the equation. 0.09375 + -0.09375 + r = -0.755836664 + -0.09375 Combine like terms: 0.09375 + -0.09375 = 0.00000 0.00000 + r = -0.755836664 + -0.09375 r = -0.755836664 + -0.09375 Combine like terms: -0.755836664 + -0.09375 = -0.849586664 r = -0.849586664 Simplifying r = -0.849586664

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.662086664, -0.849586664}

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