4r^2=11r+19

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Solution for 4r^2=11r+19 equation:


Simplifying
4r2 = 11r + 19

Reorder the terms:
4r2 = 19 + 11r

Solving
4r2 = 19 + 11r

Solving for variable 'r'.

Reorder the terms:
-19 + -11r + 4r2 = 19 + 11r + -19 + -11r

Reorder the terms:
-19 + -11r + 4r2 = 19 + -19 + 11r + -11r

Combine like terms: 19 + -19 = 0
-19 + -11r + 4r2 = 0 + 11r + -11r
-19 + -11r + 4r2 = 11r + -11r

Combine like terms: 11r + -11r = 0
-19 + -11r + 4r2 = 0

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

Divide each side by '4'.
-4.75 + -2.75r + r2 = 0

Move the constant term to the right:

Add '4.75' to each side of the equation.
-4.75 + -2.75r + 4.75 + r2 = 0 + 4.75

Reorder the terms:
-4.75 + 4.75 + -2.75r + r2 = 0 + 4.75

Combine like terms: -4.75 + 4.75 = 0.00
0.00 + -2.75r + r2 = 0 + 4.75
-2.75r + r2 = 0 + 4.75

Combine like terms: 0 + 4.75 = 4.75
-2.75r + r2 = 4.75

The r term is -2.75r.  Take half its coefficient (-1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
-2.75r + 1.890625 + r2 = 4.75 + 1.890625

Reorder the terms:
1.890625 + -2.75r + r2 = 4.75 + 1.890625

Combine like terms: 4.75 + 1.890625 = 6.640625
1.890625 + -2.75r + r2 = 6.640625

Factor a perfect square on the left side:
(r + -1.375)(r + -1.375) = 6.640625

Calculate the square root of the right side: 2.576941016

Break this problem into two subproblems by setting 
(r + -1.375) equal to 2.576941016 and -2.576941016.

Subproblem 1

r + -1.375 = 2.576941016 Simplifying r + -1.375 = 2.576941016 Reorder the terms: -1.375 + r = 2.576941016 Solving -1.375 + r = 2.576941016 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + r = 2.576941016 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + r = 2.576941016 + 1.375 r = 2.576941016 + 1.375 Combine like terms: 2.576941016 + 1.375 = 3.951941016 r = 3.951941016 Simplifying r = 3.951941016

Subproblem 2

r + -1.375 = -2.576941016 Simplifying r + -1.375 = -2.576941016 Reorder the terms: -1.375 + r = -2.576941016 Solving -1.375 + r = -2.576941016 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + r = -2.576941016 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + r = -2.576941016 + 1.375 r = -2.576941016 + 1.375 Combine like terms: -2.576941016 + 1.375 = -1.201941016 r = -1.201941016 Simplifying r = -1.201941016

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.951941016, -1.201941016}

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