20r^2=r+24

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Solution for 20r^2=r+24 equation:


Simplifying
20r2 = r + 24

Reorder the terms:
20r2 = 24 + r

Solving
20r2 = 24 + r

Solving for variable 'r'.

Reorder the terms:
-24 + -1r + 20r2 = 24 + r + -24 + -1r

Reorder the terms:
-24 + -1r + 20r2 = 24 + -24 + r + -1r

Combine like terms: 24 + -24 = 0
-24 + -1r + 20r2 = 0 + r + -1r
-24 + -1r + 20r2 = r + -1r

Combine like terms: r + -1r = 0
-24 + -1r + 20r2 = 0

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

Divide each side by '20'.
-1.2 + -0.05r + r2 = 0

Move the constant term to the right:

Add '1.2' to each side of the equation.
-1.2 + -0.05r + 1.2 + r2 = 0 + 1.2

Reorder the terms:
-1.2 + 1.2 + -0.05r + r2 = 0 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + -0.05r + r2 = 0 + 1.2
-0.05r + r2 = 0 + 1.2

Combine like terms: 0 + 1.2 = 1.2
-0.05r + r2 = 1.2

The r term is -0.05r.  Take half its coefficient (-0.025).
Square it (0.000625) and add it to both sides.

Add '0.000625' to each side of the equation.
-0.05r + 0.000625 + r2 = 1.2 + 0.000625

Reorder the terms:
0.000625 + -0.05r + r2 = 1.2 + 0.000625

Combine like terms: 1.2 + 0.000625 = 1.200625
0.000625 + -0.05r + r2 = 1.200625

Factor a perfect square on the left side:
(r + -0.025)(r + -0.025) = 1.200625

Calculate the square root of the right side: 1.09573035

Break this problem into two subproblems by setting 
(r + -0.025) equal to 1.09573035 and -1.09573035.

Subproblem 1

r + -0.025 = 1.09573035 Simplifying r + -0.025 = 1.09573035 Reorder the terms: -0.025 + r = 1.09573035 Solving -0.025 + r = 1.09573035 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.025' to each side of the equation. -0.025 + 0.025 + r = 1.09573035 + 0.025 Combine like terms: -0.025 + 0.025 = 0.000 0.000 + r = 1.09573035 + 0.025 r = 1.09573035 + 0.025 Combine like terms: 1.09573035 + 0.025 = 1.12073035 r = 1.12073035 Simplifying r = 1.12073035

Subproblem 2

r + -0.025 = -1.09573035 Simplifying r + -0.025 = -1.09573035 Reorder the terms: -0.025 + r = -1.09573035 Solving -0.025 + r = -1.09573035 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.025' to each side of the equation. -0.025 + 0.025 + r = -1.09573035 + 0.025 Combine like terms: -0.025 + 0.025 = 0.000 0.000 + r = -1.09573035 + 0.025 r = -1.09573035 + 0.025 Combine like terms: -1.09573035 + 0.025 = -1.07073035 r = -1.07073035 Simplifying r = -1.07073035

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.12073035, -1.07073035}

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