3r^2-5r-27=0

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Solution for 3r^2-5r-27=0 equation:


Simplifying
3r2 + -5r + -27 = 0

Reorder the terms:
-27 + -5r + 3r2 = 0

Solving
-27 + -5r + 3r2 = 0

Solving for variable 'r'.

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

Divide each side by '3'.
-9 + -1.666666667r + r2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + -1.666666667r + 9 + r2 = 0 + 9

Reorder the terms:
-9 + 9 + -1.666666667r + r2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -1.666666667r + r2 = 0 + 9
-1.666666667r + r2 = 0 + 9

Combine like terms: 0 + 9 = 9
-1.666666667r + r2 = 9

The r term is -1.666666667r.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667r + 0.6944444447 + r2 = 9 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667r + r2 = 9 + 0.6944444447

Combine like terms: 9 + 0.6944444447 = 9.6944444447
0.6944444447 + -1.666666667r + r2 = 9.6944444447

Factor a perfect square on the left side:
(r + -0.8333333335)(r + -0.8333333335) = 9.6944444447

Calculate the square root of the right side: 3.113590282

Break this problem into two subproblems by setting 
(r + -0.8333333335) equal to 3.113590282 and -3.113590282.

Subproblem 1

r + -0.8333333335 = 3.113590282 Simplifying r + -0.8333333335 = 3.113590282 Reorder the terms: -0.8333333335 + r = 3.113590282 Solving -0.8333333335 + r = 3.113590282 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + r = 3.113590282 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + r = 3.113590282 + 0.8333333335 r = 3.113590282 + 0.8333333335 Combine like terms: 3.113590282 + 0.8333333335 = 3.9469236155 r = 3.9469236155 Simplifying r = 3.9469236155

Subproblem 2

r + -0.8333333335 = -3.113590282 Simplifying r + -0.8333333335 = -3.113590282 Reorder the terms: -0.8333333335 + r = -3.113590282 Solving -0.8333333335 + r = -3.113590282 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + r = -3.113590282 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + r = -3.113590282 + 0.8333333335 r = -3.113590282 + 0.8333333335 Combine like terms: -3.113590282 + 0.8333333335 = -2.2802569485 r = -2.2802569485 Simplifying r = -2.2802569485

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.9469236155, -2.2802569485}

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