r^2-4r=19

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Solution for r^2-4r=19 equation:


Simplifying
r2 + -4r = 19

Reorder the terms:
-4r + r2 = 19

Solving
-4r + r2 = 19

Solving for variable 'r'.

Reorder the terms:
-19 + -4r + r2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + -4r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + -4r + 19 + r2 = 0 + 19

Reorder the terms:
-19 + 19 + -4r + r2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + -4r + r2 = 0 + 19
-4r + r2 = 0 + 19

Combine like terms: 0 + 19 = 19
-4r + r2 = 19

The r term is -4r.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4r + 4 + r2 = 19 + 4

Reorder the terms:
4 + -4r + r2 = 19 + 4

Combine like terms: 19 + 4 = 23
4 + -4r + r2 = 23

Factor a perfect square on the left side:
(r + -2)(r + -2) = 23

Calculate the square root of the right side: 4.795831523

Break this problem into two subproblems by setting 
(r + -2) equal to 4.795831523 and -4.795831523.

Subproblem 1

r + -2 = 4.795831523 Simplifying r + -2 = 4.795831523 Reorder the terms: -2 + r = 4.795831523 Solving -2 + r = 4.795831523 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + r = 4.795831523 + 2 Combine like terms: -2 + 2 = 0 0 + r = 4.795831523 + 2 r = 4.795831523 + 2 Combine like terms: 4.795831523 + 2 = 6.795831523 r = 6.795831523 Simplifying r = 6.795831523

Subproblem 2

r + -2 = -4.795831523 Simplifying r + -2 = -4.795831523 Reorder the terms: -2 + r = -4.795831523 Solving -2 + r = -4.795831523 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + r = -4.795831523 + 2 Combine like terms: -2 + 2 = 0 0 + r = -4.795831523 + 2 r = -4.795831523 + 2 Combine like terms: -4.795831523 + 2 = -2.795831523 r = -2.795831523 Simplifying r = -2.795831523

Solution

The solution to the problem is based on the solutions from the subproblems. r = {6.795831523, -2.795831523}

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