6r^2-9r=1

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Solution for 6r^2-9r=1 equation:


Simplifying
6r2 + -9r = 1

Reorder the terms:
-9r + 6r2 = 1

Solving
-9r + 6r2 = 1

Solving for variable 'r'.

Reorder the terms:
-1 + -9r + 6r2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -9r + 6r2 = 0

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

Divide each side by '6'.
-0.1666666667 + -1.5r + r2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -1.5r + 0.1666666667 + r2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -1.5r + r2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -1.5r + r2 = 0 + 0.1666666667
-1.5r + r2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-1.5r + r2 = 0.1666666667

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 = 0.1666666667 + 0.5625

Reorder the terms:
0.5625 + -1.5r + r2 = 0.1666666667 + 0.5625

Combine like terms: 0.1666666667 + 0.5625 = 0.7291666667
0.5625 + -1.5r + r2 = 0.7291666667

Factor a perfect square on the left side:
(r + -0.75)(r + -0.75) = 0.7291666667

Calculate the square root of the right side: 0.853912564

Break this problem into two subproblems by setting 
(r + -0.75) equal to 0.853912564 and -0.853912564.

Subproblem 1

r + -0.75 = 0.853912564 Simplifying r + -0.75 = 0.853912564 Reorder the terms: -0.75 + r = 0.853912564 Solving -0.75 + r = 0.853912564 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 = 0.853912564 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 0.853912564 + 0.75 r = 0.853912564 + 0.75 Combine like terms: 0.853912564 + 0.75 = 1.603912564 r = 1.603912564 Simplifying r = 1.603912564

Subproblem 2

r + -0.75 = -0.853912564 Simplifying r + -0.75 = -0.853912564 Reorder the terms: -0.75 + r = -0.853912564 Solving -0.75 + r = -0.853912564 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 = -0.853912564 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -0.853912564 + 0.75 r = -0.853912564 + 0.75 Combine like terms: -0.853912564 + 0.75 = -0.103912564 r = -0.103912564 Simplifying r = -0.103912564

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.603912564, -0.103912564}

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