r^2-40r-112=0

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Solution for r^2-40r-112=0 equation:


Simplifying
r2 + -40r + -112 = 0

Reorder the terms:
-112 + -40r + r2 = 0

Solving
-112 + -40r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '112' to each side of the equation.
-112 + -40r + 112 + r2 = 0 + 112

Reorder the terms:
-112 + 112 + -40r + r2 = 0 + 112

Combine like terms: -112 + 112 = 0
0 + -40r + r2 = 0 + 112
-40r + r2 = 0 + 112

Combine like terms: 0 + 112 = 112
-40r + r2 = 112

The r term is -40r.  Take half its coefficient (-20).
Square it (400) and add it to both sides.

Add '400' to each side of the equation.
-40r + 400 + r2 = 112 + 400

Reorder the terms:
400 + -40r + r2 = 112 + 400

Combine like terms: 112 + 400 = 512
400 + -40r + r2 = 512

Factor a perfect square on the left side:
(r + -20)(r + -20) = 512

Calculate the square root of the right side: 22.627416998

Break this problem into two subproblems by setting 
(r + -20) equal to 22.627416998 and -22.627416998.

Subproblem 1

r + -20 = 22.627416998 Simplifying r + -20 = 22.627416998 Reorder the terms: -20 + r = 22.627416998 Solving -20 + r = 22.627416998 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + r = 22.627416998 + 20 Combine like terms: -20 + 20 = 0 0 + r = 22.627416998 + 20 r = 22.627416998 + 20 Combine like terms: 22.627416998 + 20 = 42.627416998 r = 42.627416998 Simplifying r = 42.627416998

Subproblem 2

r + -20 = -22.627416998 Simplifying r + -20 = -22.627416998 Reorder the terms: -20 + r = -22.627416998 Solving -20 + r = -22.627416998 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + r = -22.627416998 + 20 Combine like terms: -20 + 20 = 0 0 + r = -22.627416998 + 20 r = -22.627416998 + 20 Combine like terms: -22.627416998 + 20 = -2.627416998 r = -2.627416998 Simplifying r = -2.627416998

Solution

The solution to the problem is based on the solutions from the subproblems. r = {42.627416998, -2.627416998}

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