-4r^2+21r=r+13

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Solution for -4r^2+21r=r+13 equation:


Simplifying
-4r2 + 21r = r + 13

Reorder the terms:
21r + -4r2 = r + 13

Reorder the terms:
21r + -4r2 = 13 + r

Solving
21r + -4r2 = 13 + r

Solving for variable 'r'.

Reorder the terms:
-13 + 21r + -1r + -4r2 = 13 + r + -13 + -1r

Combine like terms: 21r + -1r = 20r
-13 + 20r + -4r2 = 13 + r + -13 + -1r

Reorder the terms:
-13 + 20r + -4r2 = 13 + -13 + r + -1r

Combine like terms: 13 + -13 = 0
-13 + 20r + -4r2 = 0 + r + -1r
-13 + 20r + -4r2 = r + -1r

Combine like terms: r + -1r = 0
-13 + 20r + -4r2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
3.25 + -5r + r2 = 0

Move the constant term to the right:

Add '-3.25' to each side of the equation.
3.25 + -5r + -3.25 + r2 = 0 + -3.25

Reorder the terms:
3.25 + -3.25 + -5r + r2 = 0 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + -5r + r2 = 0 + -3.25
-5r + r2 = 0 + -3.25

Combine like terms: 0 + -3.25 = -3.25
-5r + r2 = -3.25

The r term is -5r.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5r + 6.25 + r2 = -3.25 + 6.25

Reorder the terms:
6.25 + -5r + r2 = -3.25 + 6.25

Combine like terms: -3.25 + 6.25 = 3
6.25 + -5r + r2 = 3

Factor a perfect square on the left side:
(r + -2.5)(r + -2.5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(r + -2.5) equal to 1.732050808 and -1.732050808.

Subproblem 1

r + -2.5 = 1.732050808 Simplifying r + -2.5 = 1.732050808 Reorder the terms: -2.5 + r = 1.732050808 Solving -2.5 + r = 1.732050808 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r = 1.732050808 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = 1.732050808 + 2.5 r = 1.732050808 + 2.5 Combine like terms: 1.732050808 + 2.5 = 4.232050808 r = 4.232050808 Simplifying r = 4.232050808

Subproblem 2

r + -2.5 = -1.732050808 Simplifying r + -2.5 = -1.732050808 Reorder the terms: -2.5 + r = -1.732050808 Solving -2.5 + r = -1.732050808 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r = -1.732050808 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = -1.732050808 + 2.5 r = -1.732050808 + 2.5 Combine like terms: -1.732050808 + 2.5 = 0.767949192 r = 0.767949192 Simplifying r = 0.767949192

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.232050808, 0.767949192}

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