2r^2-3r+4=r^2+2r+9

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Solution for 2r^2-3r+4=r^2+2r+9 equation:


Simplifying
2r2 + -3r + 4 = r2 + 2r + 9

Reorder the terms:
4 + -3r + 2r2 = r2 + 2r + 9

Reorder the terms:
4 + -3r + 2r2 = 9 + 2r + r2

Solving
4 + -3r + 2r2 = 9 + 2r + r2

Solving for variable 'r'.

Reorder the terms:
4 + -9 + -3r + -2r + 2r2 + -1r2 = 9 + 2r + r2 + -9 + -2r + -1r2

Combine like terms: 4 + -9 = -5
-5 + -3r + -2r + 2r2 + -1r2 = 9 + 2r + r2 + -9 + -2r + -1r2

Combine like terms: -3r + -2r = -5r
-5 + -5r + 2r2 + -1r2 = 9 + 2r + r2 + -9 + -2r + -1r2

Combine like terms: 2r2 + -1r2 = 1r2
-5 + -5r + 1r2 = 9 + 2r + r2 + -9 + -2r + -1r2

Reorder the terms:
-5 + -5r + 1r2 = 9 + -9 + 2r + -2r + r2 + -1r2

Combine like terms: 9 + -9 = 0
-5 + -5r + 1r2 = 0 + 2r + -2r + r2 + -1r2
-5 + -5r + 1r2 = 2r + -2r + r2 + -1r2

Combine like terms: 2r + -2r = 0
-5 + -5r + 1r2 = 0 + r2 + -1r2
-5 + -5r + 1r2 = r2 + -1r2

Combine like terms: r2 + -1r2 = 0
-5 + -5r + 1r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -5r + 5 + r2 = 0 + 5

Reorder the terms:
-5 + 5 + -5r + r2 = 0 + 5

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

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

The r term is -5r.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5r + 6.25 + r2 = 5 + 6.25

Reorder the terms:
6.25 + -5r + r2 = 5 + 6.25

Combine like terms: 5 + 6.25 = 11.25
6.25 + -5r + r2 = 11.25

Factor a perfect square on the left side:
(r + -2.5)(r + -2.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(r + -2.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

r + -2.5 = 3.354101966 Simplifying r + -2.5 = 3.354101966 Reorder the terms: -2.5 + r = 3.354101966 Solving -2.5 + r = 3.354101966 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r = 3.354101966 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = 3.354101966 + 2.5 r = 3.354101966 + 2.5 Combine like terms: 3.354101966 + 2.5 = 5.854101966 r = 5.854101966 Simplifying r = 5.854101966

Subproblem 2

r + -2.5 = -3.354101966 Simplifying r + -2.5 = -3.354101966 Reorder the terms: -2.5 + r = -3.354101966 Solving -2.5 + r = -3.354101966 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r = -3.354101966 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = -3.354101966 + 2.5 r = -3.354101966 + 2.5 Combine like terms: -3.354101966 + 2.5 = -0.854101966 r = -0.854101966 Simplifying r = -0.854101966

Solution

The solution to the problem is based on the solutions from the subproblems. r = {5.854101966, -0.854101966}

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