100r=40(r^2-6.25)

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Solution for 100r=40(r^2-6.25) equation: 


Simplifying
100r = 40(r2 + -6.25)

Reorder the terms:
100r = 40(-6.25 + r2)
100r = (-6.25 * 40 + r2 * 40)
100r = (-250 + 40r2)

Solving
100r = -250 + 40r2

Solving for variable 'r'.

Reorder the terms:
250 + 100r + -40r2 = -250 + 40r2 + 250 + -40r2

Reorder the terms:
250 + 100r + -40r2 = -250 + 250 + 40r2 + -40r2

Combine like terms: -250 + 250 = 0
250 + 100r + -40r2 = 0 + 40r2 + -40r2
250 + 100r + -40r2 = 40r2 + -40r2

Combine like terms: 40r2 + -40r2 = 0
250 + 100r + -40r2 = 0

Factor out the Greatest Common Factor (GCF), '10'.
10(25 + 10r + -4r2) = 0

Ignore the factor 10.

Subproblem 1
Set the factor '(25 + 10r + -4r2)' equal to zero and attempt to solve:

Simplifying
25 + 10r + -4r2 = 0

Solving
25 + 10r + -4r2 = 0

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

Divide each side by '-4'.
-6.25 + -2.5r + r2 = 0

Move the constant term to the right:

Add '6.25' to each side of the equation.
-6.25 + -2.5r + 6.25 + r2 = 0 + 6.25

Reorder the terms:
-6.25 + 6.25 + -2.5r + r2 = 0 + 6.25

Combine like terms: -6.25 + 6.25 = 0.00
0.00 + -2.5r + r2 = 0 + 6.25
-2.5r + r2 = 0 + 6.25

Combine like terms: 0 + 6.25 = 6.25
-2.5r + r2 = 6.25

The r term is -2.5r.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5r + 1.5625 + r2 = 6.25 + 1.5625

Reorder the terms:
1.5625 + -2.5r + r2 = 6.25 + 1.5625

Combine like terms: 6.25 + 1.5625 = 7.8125
1.5625 + -2.5r + r2 = 7.8125

Factor a perfect square on the left side:
(r + -1.25)(r + -1.25) = 7.8125

Calculate the square root of the right side: 2.795084972

Break this problem into two subproblems by setting 
(r + -1.25) equal to 2.795084972 and -2.795084972.


Subproblem 1
r + -1.25 = 2.795084972

Simplifying
r + -1.25 = 2.795084972

Reorder the terms:
-1.25 + r = 2.795084972

Solving
-1.25 + r = 2.795084972

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '1.25' to each side of the equation.
-1.25 + 1.25 + r = 2.795084972 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + r = 2.795084972 + 1.25
r = 2.795084972 + 1.25

Combine like terms: 2.795084972 + 1.25 = 4.045084972
r = 4.045084972

Simplifying
r = 4.045084972


Subproblem 2
r + -1.25 = -2.795084972

Simplifying
r + -1.25 = -2.795084972

Reorder the terms:
-1.25 + r = -2.795084972

Solving
-1.25 + r = -2.795084972

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '1.25' to each side of the equation.
-1.25 + 1.25 + r = -2.795084972 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + r = -2.795084972 + 1.25
r = -2.795084972 + 1.25

Combine like terms: -2.795084972 + 1.25 = -1.545084972
r = -1.545084972

Simplifying
r = -1.545084972


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.045084972, -1.545084972}
Solution
r = {4.045084972, -1.545084972}

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