27r=4r^2-15

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Solution for 27r=4r^2-15 equation:


Simplifying
27r = 4r2 + -15

Reorder the terms:
27r = -15 + 4r2

Solving
27r = -15 + 4r2

Solving for variable 'r'.

Reorder the terms:
15 + 27r + -4r2 = -15 + 4r2 + 15 + -4r2

Reorder the terms:
15 + 27r + -4r2 = -15 + 15 + 4r2 + -4r2

Combine like terms: -15 + 15 = 0
15 + 27r + -4r2 = 0 + 4r2 + -4r2
15 + 27r + -4r2 = 4r2 + -4r2

Combine like terms: 4r2 + -4r2 = 0
15 + 27r + -4r2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-3.75 + -6.75r + r2 = 0

Move the constant term to the right:

Add '3.75' to each side of the equation.
-3.75 + -6.75r + 3.75 + r2 = 0 + 3.75

Reorder the terms:
-3.75 + 3.75 + -6.75r + r2 = 0 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + -6.75r + r2 = 0 + 3.75
-6.75r + r2 = 0 + 3.75

Combine like terms: 0 + 3.75 = 3.75
-6.75r + r2 = 3.75

The r term is -6.75r.  Take half its coefficient (-3.375).
Square it (11.390625) and add it to both sides.

Add '11.390625' to each side of the equation.
-6.75r + 11.390625 + r2 = 3.75 + 11.390625

Reorder the terms:
11.390625 + -6.75r + r2 = 3.75 + 11.390625

Combine like terms: 3.75 + 11.390625 = 15.140625
11.390625 + -6.75r + r2 = 15.140625

Factor a perfect square on the left side:
(r + -3.375)(r + -3.375) = 15.140625

Calculate the square root of the right side: 3.891095604

Break this problem into two subproblems by setting 
(r + -3.375) equal to 3.891095604 and -3.891095604.

Subproblem 1

r + -3.375 = 3.891095604 Simplifying r + -3.375 = 3.891095604 Reorder the terms: -3.375 + r = 3.891095604 Solving -3.375 + r = 3.891095604 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.375' to each side of the equation. -3.375 + 3.375 + r = 3.891095604 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + r = 3.891095604 + 3.375 r = 3.891095604 + 3.375 Combine like terms: 3.891095604 + 3.375 = 7.266095604 r = 7.266095604 Simplifying r = 7.266095604

Subproblem 2

r + -3.375 = -3.891095604 Simplifying r + -3.375 = -3.891095604 Reorder the terms: -3.375 + r = -3.891095604 Solving -3.375 + r = -3.891095604 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.375' to each side of the equation. -3.375 + 3.375 + r = -3.891095604 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + r = -3.891095604 + 3.375 r = -3.891095604 + 3.375 Combine like terms: -3.891095604 + 3.375 = -0.516095604 r = -0.516095604 Simplifying r = -0.516095604

Solution

The solution to the problem is based on the solutions from the subproblems. r = {7.266095604, -0.516095604}

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