13r^2+24r-320=0

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Solution for 13r^2+24r-320=0 equation:


Simplifying
13r2 + 24r + -320 = 0

Reorder the terms:
-320 + 24r + 13r2 = 0

Solving
-320 + 24r + 13r2 = 0

Solving for variable 'r'.

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

Divide each side by '13'.
-24.61538462 + 1.846153846r + r2 = 0

Move the constant term to the right:

Add '24.61538462' to each side of the equation.
-24.61538462 + 1.846153846r + 24.61538462 + r2 = 0 + 24.61538462

Reorder the terms:
-24.61538462 + 24.61538462 + 1.846153846r + r2 = 0 + 24.61538462

Combine like terms: -24.61538462 + 24.61538462 = 0.00000000
0.00000000 + 1.846153846r + r2 = 0 + 24.61538462
1.846153846r + r2 = 0 + 24.61538462

Combine like terms: 0 + 24.61538462 = 24.61538462
1.846153846r + r2 = 24.61538462

The r term is 1.846153846r.  Take half its coefficient (0.923076923).
Square it (0.8520710058) and add it to both sides.

Add '0.8520710058' to each side of the equation.
1.846153846r + 0.8520710058 + r2 = 24.61538462 + 0.8520710058

Reorder the terms:
0.8520710058 + 1.846153846r + r2 = 24.61538462 + 0.8520710058

Combine like terms: 24.61538462 + 0.8520710058 = 25.4674556258
0.8520710058 + 1.846153846r + r2 = 25.4674556258

Factor a perfect square on the left side:
(r + 0.923076923)(r + 0.923076923) = 25.4674556258

Calculate the square root of the right side: 5.046529067

Break this problem into two subproblems by setting 
(r + 0.923076923) equal to 5.046529067 and -5.046529067.

Subproblem 1

r + 0.923076923 = 5.046529067 Simplifying r + 0.923076923 = 5.046529067 Reorder the terms: 0.923076923 + r = 5.046529067 Solving 0.923076923 + r = 5.046529067 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.923076923' to each side of the equation. 0.923076923 + -0.923076923 + r = 5.046529067 + -0.923076923 Combine like terms: 0.923076923 + -0.923076923 = 0.000000000 0.000000000 + r = 5.046529067 + -0.923076923 r = 5.046529067 + -0.923076923 Combine like terms: 5.046529067 + -0.923076923 = 4.123452144 r = 4.123452144 Simplifying r = 4.123452144

Subproblem 2

r + 0.923076923 = -5.046529067 Simplifying r + 0.923076923 = -5.046529067 Reorder the terms: 0.923076923 + r = -5.046529067 Solving 0.923076923 + r = -5.046529067 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.923076923' to each side of the equation. 0.923076923 + -0.923076923 + r = -5.046529067 + -0.923076923 Combine like terms: 0.923076923 + -0.923076923 = 0.000000000 0.000000000 + r = -5.046529067 + -0.923076923 r = -5.046529067 + -0.923076923 Combine like terms: -5.046529067 + -0.923076923 = -5.96960599 r = -5.96960599 Simplifying r = -5.96960599

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.123452144, -5.96960599}

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