(r-5)(r+9)=-31

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Solution for (r-5)(r+9)=-31 equation:


Simplifying
(r + -5)(r + 9) = -31

Reorder the terms:
(-5 + r)(r + 9) = -31

Reorder the terms:
(-5 + r)(9 + r) = -31

Multiply (-5 + r) * (9 + r)
(-5(9 + r) + r(9 + r)) = -31
((9 * -5 + r * -5) + r(9 + r)) = -31
((-45 + -5r) + r(9 + r)) = -31
(-45 + -5r + (9 * r + r * r)) = -31
(-45 + -5r + (9r + r2)) = -31

Combine like terms: -5r + 9r = 4r
(-45 + 4r + r2) = -31

Solving
-45 + 4r + r2 = -31

Solving for variable 'r'.

Reorder the terms:
-45 + 31 + 4r + r2 = -31 + 31

Combine like terms: -45 + 31 = -14
-14 + 4r + r2 = -31 + 31

Combine like terms: -31 + 31 = 0
-14 + 4r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 4r + 14 + r2 = 0 + 14

Reorder the terms:
-14 + 14 + 4r + r2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 4r + r2 = 0 + 14
4r + r2 = 0 + 14

Combine like terms: 0 + 14 = 14
4r + r2 = 14

The r term is 4r.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4r + 4 + r2 = 14 + 4

Reorder the terms:
4 + 4r + r2 = 14 + 4

Combine like terms: 14 + 4 = 18
4 + 4r + r2 = 18

Factor a perfect square on the left side:
(r + 2)(r + 2) = 18

Calculate the square root of the right side: 4.242640687

Break this problem into two subproblems by setting 
(r + 2) equal to 4.242640687 and -4.242640687.

Subproblem 1

r + 2 = 4.242640687 Simplifying r + 2 = 4.242640687 Reorder the terms: 2 + r = 4.242640687 Solving 2 + r = 4.242640687 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = 4.242640687 + -2 Combine like terms: 2 + -2 = 0 0 + r = 4.242640687 + -2 r = 4.242640687 + -2 Combine like terms: 4.242640687 + -2 = 2.242640687 r = 2.242640687 Simplifying r = 2.242640687

Subproblem 2

r + 2 = -4.242640687 Simplifying r + 2 = -4.242640687 Reorder the terms: 2 + r = -4.242640687 Solving 2 + r = -4.242640687 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = -4.242640687 + -2 Combine like terms: 2 + -2 = 0 0 + r = -4.242640687 + -2 r = -4.242640687 + -2 Combine like terms: -4.242640687 + -2 = -6.242640687 r = -6.242640687 Simplifying r = -6.242640687

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.242640687, -6.242640687}

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