5r^2-13r+19=3r^2

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Solution for 5r^2-13r+19=3r^2 equation:


Simplifying
5r2 + -13r + 19 = 3r2

Reorder the terms:
19 + -13r + 5r2 = 3r2

Solving
19 + -13r + 5r2 = 3r2

Solving for variable 'r'.

Combine like terms: 5r2 + -3r2 = 2r2
19 + -13r + 2r2 = 3r2 + -3r2

Combine like terms: 3r2 + -3r2 = 0
19 + -13r + 2r2 = 0

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

Divide each side by '2'.
9.5 + -6.5r + r2 = 0

Move the constant term to the right:

Add '-9.5' to each side of the equation.
9.5 + -6.5r + -9.5 + r2 = 0 + -9.5

Reorder the terms:
9.5 + -9.5 + -6.5r + r2 = 0 + -9.5

Combine like terms: 9.5 + -9.5 = 0.0
0.0 + -6.5r + r2 = 0 + -9.5
-6.5r + r2 = 0 + -9.5

Combine like terms: 0 + -9.5 = -9.5
-6.5r + r2 = -9.5

The r term is -6.5r.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5r + 10.5625 + r2 = -9.5 + 10.5625

Reorder the terms:
10.5625 + -6.5r + r2 = -9.5 + 10.5625

Combine like terms: -9.5 + 10.5625 = 1.0625
10.5625 + -6.5r + r2 = 1.0625

Factor a perfect square on the left side:
(r + -3.25)(r + -3.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(r + -3.25) equal to 1.030776406 and -1.030776406.

Subproblem 1

r + -3.25 = 1.030776406 Simplifying r + -3.25 = 1.030776406 Reorder the terms: -3.25 + r = 1.030776406 Solving -3.25 + r = 1.030776406 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + r = 1.030776406 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + r = 1.030776406 + 3.25 r = 1.030776406 + 3.25 Combine like terms: 1.030776406 + 3.25 = 4.280776406 r = 4.280776406 Simplifying r = 4.280776406

Subproblem 2

r + -3.25 = -1.030776406 Simplifying r + -3.25 = -1.030776406 Reorder the terms: -3.25 + r = -1.030776406 Solving -3.25 + r = -1.030776406 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + r = -1.030776406 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + r = -1.030776406 + 3.25 r = -1.030776406 + 3.25 Combine like terms: -1.030776406 + 3.25 = 2.219223594 r = 2.219223594 Simplifying r = 2.219223594

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.280776406, 2.219223594}

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