4r^2=12r+5

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Solution for 4r^2=12r+5 equation:


Simplifying
4r2 = 12r + 5

Reorder the terms:
4r2 = 5 + 12r

Solving
4r2 = 5 + 12r

Solving for variable 'r'.

Reorder the terms:
-5 + -12r + 4r2 = 5 + 12r + -5 + -12r

Reorder the terms:
-5 + -12r + 4r2 = 5 + -5 + 12r + -12r

Combine like terms: 5 + -5 = 0
-5 + -12r + 4r2 = 0 + 12r + -12r
-5 + -12r + 4r2 = 12r + -12r

Combine like terms: 12r + -12r = 0
-5 + -12r + 4r2 = 0

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

Divide each side by '4'.
-1.25 + -3r + r2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -3r + 1.25 + r2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -3r + r2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -3r + r2 = 0 + 1.25
-3r + r2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-3r + r2 = 1.25

The r term is -3r.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3r + 2.25 + r2 = 1.25 + 2.25

Reorder the terms:
2.25 + -3r + r2 = 1.25 + 2.25

Combine like terms: 1.25 + 2.25 = 3.5
2.25 + -3r + r2 = 3.5

Factor a perfect square on the left side:
(r + -1.5)(r + -1.5) = 3.5

Calculate the square root of the right side: 1.870828693

Break this problem into two subproblems by setting 
(r + -1.5) equal to 1.870828693 and -1.870828693.

Subproblem 1

r + -1.5 = 1.870828693 Simplifying r + -1.5 = 1.870828693 Reorder the terms: -1.5 + r = 1.870828693 Solving -1.5 + r = 1.870828693 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + r = 1.870828693 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = 1.870828693 + 1.5 r = 1.870828693 + 1.5 Combine like terms: 1.870828693 + 1.5 = 3.370828693 r = 3.370828693 Simplifying r = 3.370828693

Subproblem 2

r + -1.5 = -1.870828693 Simplifying r + -1.5 = -1.870828693 Reorder the terms: -1.5 + r = -1.870828693 Solving -1.5 + r = -1.870828693 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + r = -1.870828693 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = -1.870828693 + 1.5 r = -1.870828693 + 1.5 Combine like terms: -1.870828693 + 1.5 = -0.370828693 r = -0.370828693 Simplifying r = -0.370828693

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.370828693, -0.370828693}

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