(r-3)(r+7)=23

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Solution for (r-3)(r+7)=23 equation:


Simplifying
(r + -3)(r + 7) = 23

Reorder the terms:
(-3 + r)(r + 7) = 23

Reorder the terms:
(-3 + r)(7 + r) = 23

Multiply (-3 + r) * (7 + r)
(-3(7 + r) + r(7 + r)) = 23
((7 * -3 + r * -3) + r(7 + r)) = 23
((-21 + -3r) + r(7 + r)) = 23
(-21 + -3r + (7 * r + r * r)) = 23
(-21 + -3r + (7r + r2)) = 23

Combine like terms: -3r + 7r = 4r
(-21 + 4r + r2) = 23

Solving
-21 + 4r + r2 = 23

Solving for variable 'r'.

Reorder the terms:
-21 + -23 + 4r + r2 = 23 + -23

Combine like terms: -21 + -23 = -44
-44 + 4r + r2 = 23 + -23

Combine like terms: 23 + -23 = 0
-44 + 4r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '44' to each side of the equation.
-44 + 4r + 44 + r2 = 0 + 44

Reorder the terms:
-44 + 44 + 4r + r2 = 0 + 44

Combine like terms: -44 + 44 = 0
0 + 4r + r2 = 0 + 44
4r + r2 = 0 + 44

Combine like terms: 0 + 44 = 44
4r + r2 = 44

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 = 44 + 4

Reorder the terms:
4 + 4r + r2 = 44 + 4

Combine like terms: 44 + 4 = 48
4 + 4r + r2 = 48

Factor a perfect square on the left side:
(r + 2)(r + 2) = 48

Calculate the square root of the right side: 6.92820323

Break this problem into two subproblems by setting 
(r + 2) equal to 6.92820323 and -6.92820323.

Subproblem 1

r + 2 = 6.92820323 Simplifying r + 2 = 6.92820323 Reorder the terms: 2 + r = 6.92820323 Solving 2 + r = 6.92820323 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 = 6.92820323 + -2 Combine like terms: 2 + -2 = 0 0 + r = 6.92820323 + -2 r = 6.92820323 + -2 Combine like terms: 6.92820323 + -2 = 4.92820323 r = 4.92820323 Simplifying r = 4.92820323

Subproblem 2

r + 2 = -6.92820323 Simplifying r + 2 = -6.92820323 Reorder the terms: 2 + r = -6.92820323 Solving 2 + r = -6.92820323 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 = -6.92820323 + -2 Combine like terms: 2 + -2 = 0 0 + r = -6.92820323 + -2 r = -6.92820323 + -2 Combine like terms: -6.92820323 + -2 = -8.92820323 r = -8.92820323 Simplifying r = -8.92820323

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.92820323, -8.92820323}

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