(r^2+5)-(3r^2-5r)=

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Solution for (r^2+5)-(3r^2-5r)= equation:


Simplifying
(r2 + 5) + -1(3r2 + -5r) = 0

Reorder the terms:
(5 + r2) + -1(3r2 + -5r) = 0

Remove parenthesis around (5 + r2)
5 + r2 + -1(3r2 + -5r) = 0

Reorder the terms:
5 + r2 + -1(-5r + 3r2) = 0
5 + r2 + (-5r * -1 + 3r2 * -1) = 0
5 + r2 + (5r + -3r2) = 0

Reorder the terms:
5 + 5r + r2 + -3r2 = 0

Combine like terms: r2 + -3r2 = -2r2
5 + 5r + -2r2 = 0

Solving
5 + 5r + -2r2 = 0

Solving for variable 'r'.

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

Divide each side by '-2'.
-2.5 + -2.5r + r2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -2.5r + 2.5 + r2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -2.5r + r2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -2.5r + r2 = 0 + 2.5
-2.5r + r2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-2.5r + r2 = 2.5

The r term is -2.5r.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5r + 1.5625 + r2 = 2.5 + 1.5625

Reorder the terms:
1.5625 + -2.5r + r2 = 2.5 + 1.5625

Combine like terms: 2.5 + 1.5625 = 4.0625
1.5625 + -2.5r + r2 = 4.0625

Factor a perfect square on the left side:
(r + -1.25)(r + -1.25) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(r + -1.25) equal to 2.015564437 and -2.015564437.

Subproblem 1

r + -1.25 = 2.015564437 Simplifying r + -1.25 = 2.015564437 Reorder the terms: -1.25 + r = 2.015564437 Solving -1.25 + r = 2.015564437 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + r = 2.015564437 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + r = 2.015564437 + 1.25 r = 2.015564437 + 1.25 Combine like terms: 2.015564437 + 1.25 = 3.265564437 r = 3.265564437 Simplifying r = 3.265564437

Subproblem 2

r + -1.25 = -2.015564437 Simplifying r + -1.25 = -2.015564437 Reorder the terms: -1.25 + r = -2.015564437 Solving -1.25 + r = -2.015564437 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + r = -2.015564437 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + r = -2.015564437 + 1.25 r = -2.015564437 + 1.25 Combine like terms: -2.015564437 + 1.25 = -0.765564437 r = -0.765564437 Simplifying r = -0.765564437

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.265564437, -0.765564437}

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