24r^2-48r-245=0

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Solution for 24r^2-48r-245=0 equation:


Simplifying
24r2 + -48r + -245 = 0

Reorder the terms:
-245 + -48r + 24r2 = 0

Solving
-245 + -48r + 24r2 = 0

Solving for variable 'r'.

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

Divide each side by '24'.
-10.20833333 + -2r + r2 = 0

Move the constant term to the right:

Add '10.20833333' to each side of the equation.
-10.20833333 + -2r + 10.20833333 + r2 = 0 + 10.20833333

Reorder the terms:
-10.20833333 + 10.20833333 + -2r + r2 = 0 + 10.20833333

Combine like terms: -10.20833333 + 10.20833333 = 0.00000000
0.00000000 + -2r + r2 = 0 + 10.20833333
-2r + r2 = 0 + 10.20833333

Combine like terms: 0 + 10.20833333 = 10.20833333
-2r + r2 = 10.20833333

The r term is -2r.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2r + 1 + r2 = 10.20833333 + 1

Reorder the terms:
1 + -2r + r2 = 10.20833333 + 1

Combine like terms: 10.20833333 + 1 = 11.20833333
1 + -2r + r2 = 11.20833333

Factor a perfect square on the left side:
(r + -1)(r + -1) = 11.20833333

Calculate the square root of the right side: 3.347884904

Break this problem into two subproblems by setting 
(r + -1) equal to 3.347884904 and -3.347884904.

Subproblem 1

r + -1 = 3.347884904 Simplifying r + -1 = 3.347884904 Reorder the terms: -1 + r = 3.347884904 Solving -1 + r = 3.347884904 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = 3.347884904 + 1 Combine like terms: -1 + 1 = 0 0 + r = 3.347884904 + 1 r = 3.347884904 + 1 Combine like terms: 3.347884904 + 1 = 4.347884904 r = 4.347884904 Simplifying r = 4.347884904

Subproblem 2

r + -1 = -3.347884904 Simplifying r + -1 = -3.347884904 Reorder the terms: -1 + r = -3.347884904 Solving -1 + r = -3.347884904 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = -3.347884904 + 1 Combine like terms: -1 + 1 = 0 0 + r = -3.347884904 + 1 r = -3.347884904 + 1 Combine like terms: -3.347884904 + 1 = -2.347884904 r = -2.347884904 Simplifying r = -2.347884904

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.347884904, -2.347884904}

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