(r-4)(r+10)=-22

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Solution for (r-4)(r+10)=-22 equation:


Simplifying
(r + -4)(r + 10) = -22

Reorder the terms:
(-4 + r)(r + 10) = -22

Reorder the terms:
(-4 + r)(10 + r) = -22

Multiply (-4 + r) * (10 + r)
(-4(10 + r) + r(10 + r)) = -22
((10 * -4 + r * -4) + r(10 + r)) = -22
((-40 + -4r) + r(10 + r)) = -22
(-40 + -4r + (10 * r + r * r)) = -22
(-40 + -4r + (10r + r2)) = -22

Combine like terms: -4r + 10r = 6r
(-40 + 6r + r2) = -22

Solving
-40 + 6r + r2 = -22

Solving for variable 'r'.

Reorder the terms:
-40 + 22 + 6r + r2 = -22 + 22

Combine like terms: -40 + 22 = -18
-18 + 6r + r2 = -22 + 22

Combine like terms: -22 + 22 = 0
-18 + 6r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 6r + 18 + r2 = 0 + 18

Reorder the terms:
-18 + 18 + 6r + r2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + 6r + r2 = 0 + 18
6r + r2 = 0 + 18

Combine like terms: 0 + 18 = 18
6r + r2 = 18

The r term is 6r.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6r + 9 + r2 = 18 + 9

Reorder the terms:
9 + 6r + r2 = 18 + 9

Combine like terms: 18 + 9 = 27
9 + 6r + r2 = 27

Factor a perfect square on the left side:
(r + 3)(r + 3) = 27

Calculate the square root of the right side: 5.196152423

Break this problem into two subproblems by setting 
(r + 3) equal to 5.196152423 and -5.196152423.

Subproblem 1

r + 3 = 5.196152423 Simplifying r + 3 = 5.196152423 Reorder the terms: 3 + r = 5.196152423 Solving 3 + r = 5.196152423 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = 5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + r = 5.196152423 + -3 r = 5.196152423 + -3 Combine like terms: 5.196152423 + -3 = 2.196152423 r = 2.196152423 Simplifying r = 2.196152423

Subproblem 2

r + 3 = -5.196152423 Simplifying r + 3 = -5.196152423 Reorder the terms: 3 + r = -5.196152423 Solving 3 + r = -5.196152423 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = -5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + r = -5.196152423 + -3 r = -5.196152423 + -3 Combine like terms: -5.196152423 + -3 = -8.196152423 r = -8.196152423 Simplifying r = -8.196152423

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.196152423, -8.196152423}

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