19r^2+20r-475=0

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Solution for 19r^2+20r-475=0 equation:


Simplifying
19r2 + 20r + -475 = 0

Reorder the terms:
-475 + 20r + 19r2 = 0

Solving
-475 + 20r + 19r2 = 0

Solving for variable 'r'.

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

Divide each side by '19'.
-25 + 1.052631579r + r2 = 0

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 1.052631579r + 25 + r2 = 0 + 25

Reorder the terms:
-25 + 25 + 1.052631579r + r2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 1.052631579r + r2 = 0 + 25
1.052631579r + r2 = 0 + 25

Combine like terms: 0 + 25 = 25
1.052631579r + r2 = 25

The r term is 1.052631579r.  Take half its coefficient (0.5263157895).
Square it (0.2770083103) and add it to both sides.

Add '0.2770083103' to each side of the equation.
1.052631579r + 0.2770083103 + r2 = 25 + 0.2770083103

Reorder the terms:
0.2770083103 + 1.052631579r + r2 = 25 + 0.2770083103

Combine like terms: 25 + 0.2770083103 = 25.2770083103
0.2770083103 + 1.052631579r + r2 = 25.2770083103

Factor a perfect square on the left side:
(r + 0.5263157895)(r + 0.5263157895) = 25.2770083103

Calculate the square root of the right side: 5.02762452

Break this problem into two subproblems by setting 
(r + 0.5263157895) equal to 5.02762452 and -5.02762452.

Subproblem 1

r + 0.5263157895 = 5.02762452 Simplifying r + 0.5263157895 = 5.02762452 Reorder the terms: 0.5263157895 + r = 5.02762452 Solving 0.5263157895 + r = 5.02762452 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5263157895' to each side of the equation. 0.5263157895 + -0.5263157895 + r = 5.02762452 + -0.5263157895 Combine like terms: 0.5263157895 + -0.5263157895 = 0.0000000000 0.0000000000 + r = 5.02762452 + -0.5263157895 r = 5.02762452 + -0.5263157895 Combine like terms: 5.02762452 + -0.5263157895 = 4.5013087305 r = 4.5013087305 Simplifying r = 4.5013087305

Subproblem 2

r + 0.5263157895 = -5.02762452 Simplifying r + 0.5263157895 = -5.02762452 Reorder the terms: 0.5263157895 + r = -5.02762452 Solving 0.5263157895 + r = -5.02762452 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5263157895' to each side of the equation. 0.5263157895 + -0.5263157895 + r = -5.02762452 + -0.5263157895 Combine like terms: 0.5263157895 + -0.5263157895 = 0.0000000000 0.0000000000 + r = -5.02762452 + -0.5263157895 r = -5.02762452 + -0.5263157895 Combine like terms: -5.02762452 + -0.5263157895 = -5.5539403095 r = -5.5539403095 Simplifying r = -5.5539403095

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.5013087305, -5.5539403095}

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