16r^2=6+24r-9

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Solution for 16r^2=6+24r-9 equation:


Simplifying
16r2 = 6 + 24r + -9

Reorder the terms:
16r2 = 6 + -9 + 24r

Combine like terms: 6 + -9 = -3
16r2 = -3 + 24r

Solving
16r2 = -3 + 24r

Solving for variable 'r'.

Reorder the terms:
3 + -24r + 16r2 = -3 + 24r + 3 + -24r

Reorder the terms:
3 + -24r + 16r2 = -3 + 3 + 24r + -24r

Combine like terms: -3 + 3 = 0
3 + -24r + 16r2 = 0 + 24r + -24r
3 + -24r + 16r2 = 24r + -24r

Combine like terms: 24r + -24r = 0
3 + -24r + 16r2 = 0

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

Divide each side by '16'.
0.1875 + -1.5r + r2 = 0

Move the constant term to the right:

Add '-0.1875' to each side of the equation.
0.1875 + -1.5r + -0.1875 + r2 = 0 + -0.1875

Reorder the terms:
0.1875 + -0.1875 + -1.5r + r2 = 0 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + -1.5r + r2 = 0 + -0.1875
-1.5r + r2 = 0 + -0.1875

Combine like terms: 0 + -0.1875 = -0.1875
-1.5r + r2 = -0.1875

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 = -0.1875 + 0.5625

Reorder the terms:
0.5625 + -1.5r + r2 = -0.1875 + 0.5625

Combine like terms: -0.1875 + 0.5625 = 0.375
0.5625 + -1.5r + r2 = 0.375

Factor a perfect square on the left side:
(r + -0.75)(r + -0.75) = 0.375

Calculate the square root of the right side: 0.612372436

Break this problem into two subproblems by setting 
(r + -0.75) equal to 0.612372436 and -0.612372436.

Subproblem 1

r + -0.75 = 0.612372436 Simplifying r + -0.75 = 0.612372436 Reorder the terms: -0.75 + r = 0.612372436 Solving -0.75 + r = 0.612372436 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 = 0.612372436 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 0.612372436 + 0.75 r = 0.612372436 + 0.75 Combine like terms: 0.612372436 + 0.75 = 1.362372436 r = 1.362372436 Simplifying r = 1.362372436

Subproblem 2

r + -0.75 = -0.612372436 Simplifying r + -0.75 = -0.612372436 Reorder the terms: -0.75 + r = -0.612372436 Solving -0.75 + r = -0.612372436 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 = -0.612372436 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -0.612372436 + 0.75 r = -0.612372436 + 0.75 Combine like terms: -0.612372436 + 0.75 = 0.137627564 r = 0.137627564 Simplifying r = 0.137627564

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.362372436, 0.137627564}

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