2r^2+7r-33=0

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Solution for 2r^2+7r-33=0 equation:


Simplifying
2r2 + 7r + -33 = 0

Reorder the terms:
-33 + 7r + 2r2 = 0

Solving
-33 + 7r + 2r2 = 0

Solving for variable 'r'.

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

Divide each side by '2'.
-16.5 + 3.5r + r2 = 0

Move the constant term to the right:

Add '16.5' to each side of the equation.
-16.5 + 3.5r + 16.5 + r2 = 0 + 16.5

Reorder the terms:
-16.5 + 16.5 + 3.5r + r2 = 0 + 16.5

Combine like terms: -16.5 + 16.5 = 0.0
0.0 + 3.5r + r2 = 0 + 16.5
3.5r + r2 = 0 + 16.5

Combine like terms: 0 + 16.5 = 16.5
3.5r + r2 = 16.5

The r term is 3.5r.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5r + 3.0625 + r2 = 16.5 + 3.0625

Reorder the terms:
3.0625 + 3.5r + r2 = 16.5 + 3.0625

Combine like terms: 16.5 + 3.0625 = 19.5625
3.0625 + 3.5r + r2 = 19.5625

Factor a perfect square on the left side:
(r + 1.75)(r + 1.75) = 19.5625

Calculate the square root of the right side: 4.422951503

Break this problem into two subproblems by setting 
(r + 1.75) equal to 4.422951503 and -4.422951503.

Subproblem 1

r + 1.75 = 4.422951503 Simplifying r + 1.75 = 4.422951503 Reorder the terms: 1.75 + r = 4.422951503 Solving 1.75 + r = 4.422951503 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + r = 4.422951503 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + r = 4.422951503 + -1.75 r = 4.422951503 + -1.75 Combine like terms: 4.422951503 + -1.75 = 2.672951503 r = 2.672951503 Simplifying r = 2.672951503

Subproblem 2

r + 1.75 = -4.422951503 Simplifying r + 1.75 = -4.422951503 Reorder the terms: 1.75 + r = -4.422951503 Solving 1.75 + r = -4.422951503 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + r = -4.422951503 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + r = -4.422951503 + -1.75 r = -4.422951503 + -1.75 Combine like terms: -4.422951503 + -1.75 = -6.172951503 r = -6.172951503 Simplifying r = -6.172951503

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.672951503, -6.172951503}

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