5(3r-4)-(r^2+2)-2r(r-3)=

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Solution for 5(3r-4)-(r^2+2)-2r(r-3)= equation:


Simplifying
5(3r + -4) + -1(r2 + 2) + -2r(r + -3) = 0

Reorder the terms:
5(-4 + 3r) + -1(r2 + 2) + -2r(r + -3) = 0
(-4 * 5 + 3r * 5) + -1(r2 + 2) + -2r(r + -3) = 0
(-20 + 15r) + -1(r2 + 2) + -2r(r + -3) = 0

Reorder the terms:
-20 + 15r + -1(2 + r2) + -2r(r + -3) = 0
-20 + 15r + (2 * -1 + r2 * -1) + -2r(r + -3) = 0
-20 + 15r + (-2 + -1r2) + -2r(r + -3) = 0

Reorder the terms:
-20 + 15r + -2 + -1r2 + -2r(-3 + r) = 0
-20 + 15r + -2 + -1r2 + (-3 * -2r + r * -2r) = 0
-20 + 15r + -2 + -1r2 + (6r + -2r2) = 0

Reorder the terms:
-20 + -2 + 15r + 6r + -1r2 + -2r2 = 0

Combine like terms: -20 + -2 = -22
-22 + 15r + 6r + -1r2 + -2r2 = 0

Combine like terms: 15r + 6r = 21r
-22 + 21r + -1r2 + -2r2 = 0

Combine like terms: -1r2 + -2r2 = -3r2
-22 + 21r + -3r2 = 0

Solving
-22 + 21r + -3r2 = 0

Solving for variable 'r'.

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
7.333333333 + -7r + r2 = 0

Move the constant term to the right:

Add '-7.333333333' to each side of the equation.
7.333333333 + -7r + -7.333333333 + r2 = 0 + -7.333333333

Reorder the terms:
7.333333333 + -7.333333333 + -7r + r2 = 0 + -7.333333333

Combine like terms: 7.333333333 + -7.333333333 = 0.000000000
0.000000000 + -7r + r2 = 0 + -7.333333333
-7r + r2 = 0 + -7.333333333

Combine like terms: 0 + -7.333333333 = -7.333333333
-7r + r2 = -7.333333333

The r term is -7r.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7r + 12.25 + r2 = -7.333333333 + 12.25

Reorder the terms:
12.25 + -7r + r2 = -7.333333333 + 12.25

Combine like terms: -7.333333333 + 12.25 = 4.916666667
12.25 + -7r + r2 = 4.916666667

Factor a perfect square on the left side:
(r + -3.5)(r + -3.5) = 4.916666667

Calculate the square root of the right side: 2.217355783

Break this problem into two subproblems by setting 
(r + -3.5) equal to 2.217355783 and -2.217355783.

Subproblem 1

r + -3.5 = 2.217355783 Simplifying r + -3.5 = 2.217355783 Reorder the terms: -3.5 + r = 2.217355783 Solving -3.5 + r = 2.217355783 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + r = 2.217355783 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + r = 2.217355783 + 3.5 r = 2.217355783 + 3.5 Combine like terms: 2.217355783 + 3.5 = 5.717355783 r = 5.717355783 Simplifying r = 5.717355783

Subproblem 2

r + -3.5 = -2.217355783 Simplifying r + -3.5 = -2.217355783 Reorder the terms: -3.5 + r = -2.217355783 Solving -3.5 + r = -2.217355783 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + r = -2.217355783 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + r = -2.217355783 + 3.5 r = -2.217355783 + 3.5 Combine like terms: -2.217355783 + 3.5 = 1.282644217 r = 1.282644217 Simplifying r = 1.282644217

Solution

The solution to the problem is based on the solutions from the subproblems. r = {5.717355783, 1.282644217}

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