3r^2-5r-33=0

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Solution for 3r^2-5r-33=0 equation:


Simplifying
3r2 + -5r + -33 = 0

Reorder the terms:
-33 + -5r + 3r2 = 0

Solving
-33 + -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'.
-11 + -1.666666667r + r2 = 0

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + -1.666666667r + 11 + r2 = 0 + 11

Reorder the terms:
-11 + 11 + -1.666666667r + r2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + -1.666666667r + r2 = 0 + 11
-1.666666667r + r2 = 0 + 11

Combine like terms: 0 + 11 = 11
-1.666666667r + r2 = 11

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 = 11 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667r + r2 = 11 + 0.6944444447

Combine like terms: 11 + 0.6944444447 = 11.6944444447
0.6944444447 + -1.666666667r + r2 = 11.6944444447

Factor a perfect square on the left side:
(r + -0.8333333335)(r + -0.8333333335) = 11.6944444447

Calculate the square root of the right side: 3.419714088

Break this problem into two subproblems by setting 
(r + -0.8333333335) equal to 3.419714088 and -3.419714088.

Subproblem 1

r + -0.8333333335 = 3.419714088 Simplifying r + -0.8333333335 = 3.419714088 Reorder the terms: -0.8333333335 + r = 3.419714088 Solving -0.8333333335 + r = 3.419714088 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 = 3.419714088 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + r = 3.419714088 + 0.8333333335 r = 3.419714088 + 0.8333333335 Combine like terms: 3.419714088 + 0.8333333335 = 4.2530474215 r = 4.2530474215 Simplifying r = 4.2530474215

Subproblem 2

r + -0.8333333335 = -3.419714088 Simplifying r + -0.8333333335 = -3.419714088 Reorder the terms: -0.8333333335 + r = -3.419714088 Solving -0.8333333335 + r = -3.419714088 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 = -3.419714088 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + r = -3.419714088 + 0.8333333335 r = -3.419714088 + 0.8333333335 Combine like terms: -3.419714088 + 0.8333333335 = -2.5863807545 r = -2.5863807545 Simplifying r = -2.5863807545

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.2530474215, -2.5863807545}

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