(r-3)(r+7)=-5

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Solution for (r-3)(r+7)=-5 equation:


Simplifying
(r + -3)(r + 7) = -5

Reorder the terms:
(-3 + r)(r + 7) = -5

Reorder the terms:
(-3 + r)(7 + r) = -5

Multiply (-3 + r) * (7 + r)
(-3(7 + r) + r(7 + r)) = -5
((7 * -3 + r * -3) + r(7 + r)) = -5
((-21 + -3r) + r(7 + r)) = -5
(-21 + -3r + (7 * r + r * r)) = -5
(-21 + -3r + (7r + r2)) = -5

Combine like terms: -3r + 7r = 4r
(-21 + 4r + r2) = -5

Solving
-21 + 4r + r2 = -5

Solving for variable 'r'.

Reorder the terms:
-21 + 5 + 4r + r2 = -5 + 5

Combine like terms: -21 + 5 = -16
-16 + 4r + r2 = -5 + 5

Combine like terms: -5 + 5 = 0
-16 + 4r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 4r + 16 + r2 = 0 + 16

Reorder the terms:
-16 + 16 + 4r + r2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 4r + r2 = 0 + 16
4r + r2 = 0 + 16

Combine like terms: 0 + 16 = 16
4r + r2 = 16

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 = 16 + 4

Reorder the terms:
4 + 4r + r2 = 16 + 4

Combine like terms: 16 + 4 = 20
4 + 4r + r2 = 20

Factor a perfect square on the left side:
(r + 2)(r + 2) = 20

Calculate the square root of the right side: 4.472135955

Break this problem into two subproblems by setting 
(r + 2) equal to 4.472135955 and -4.472135955.

Subproblem 1

r + 2 = 4.472135955 Simplifying r + 2 = 4.472135955 Reorder the terms: 2 + r = 4.472135955 Solving 2 + r = 4.472135955 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 = 4.472135955 + -2 Combine like terms: 2 + -2 = 0 0 + r = 4.472135955 + -2 r = 4.472135955 + -2 Combine like terms: 4.472135955 + -2 = 2.472135955 r = 2.472135955 Simplifying r = 2.472135955

Subproblem 2

r + 2 = -4.472135955 Simplifying r + 2 = -4.472135955 Reorder the terms: 2 + r = -4.472135955 Solving 2 + r = -4.472135955 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 = -4.472135955 + -2 Combine like terms: 2 + -2 = 0 0 + r = -4.472135955 + -2 r = -4.472135955 + -2 Combine like terms: -4.472135955 + -2 = -6.472135955 r = -6.472135955 Simplifying r = -6.472135955

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.472135955, -6.472135955}

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