2r^2-9r-4=0

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Solution for 2r^2-9r-4=0 equation:


Simplifying
2r2 + -9r + -4 = 0

Reorder the terms:
-4 + -9r + 2r2 = 0

Solving
-4 + -9r + 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'.
-2 + -4.5r + r2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -4.5r + 2 + r2 = 0 + 2

Reorder the terms:
-2 + 2 + -4.5r + r2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -4.5r + r2 = 0 + 2
-4.5r + r2 = 0 + 2

Combine like terms: 0 + 2 = 2
-4.5r + r2 = 2

The r term is -4.5r.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5r + 5.0625 + r2 = 2 + 5.0625

Reorder the terms:
5.0625 + -4.5r + r2 = 2 + 5.0625

Combine like terms: 2 + 5.0625 = 7.0625
5.0625 + -4.5r + r2 = 7.0625

Factor a perfect square on the left side:
(r + -2.25)(r + -2.25) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(r + -2.25) equal to 2.657536453 and -2.657536453.

Subproblem 1

r + -2.25 = 2.657536453 Simplifying r + -2.25 = 2.657536453 Reorder the terms: -2.25 + r = 2.657536453 Solving -2.25 + r = 2.657536453 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + r = 2.657536453 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + r = 2.657536453 + 2.25 r = 2.657536453 + 2.25 Combine like terms: 2.657536453 + 2.25 = 4.907536453 r = 4.907536453 Simplifying r = 4.907536453

Subproblem 2

r + -2.25 = -2.657536453 Simplifying r + -2.25 = -2.657536453 Reorder the terms: -2.25 + r = -2.657536453 Solving -2.25 + r = -2.657536453 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + r = -2.657536453 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + r = -2.657536453 + 2.25 r = -2.657536453 + 2.25 Combine like terms: -2.657536453 + 2.25 = -0.407536453 r = -0.407536453 Simplifying r = -0.407536453

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.907536453, -0.407536453}

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