r^2+3r-112=0

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Solution for r^2+3r-112=0 equation:


Simplifying
r2 + 3r + -112 = 0

Reorder the terms:
-112 + 3r + r2 = 0

Solving
-112 + 3r + 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 + 3r + 112 + r2 = 0 + 112

Reorder the terms:
-112 + 112 + 3r + r2 = 0 + 112

Combine like terms: -112 + 112 = 0
0 + 3r + r2 = 0 + 112
3r + r2 = 0 + 112

Combine like terms: 0 + 112 = 112
3r + r2 = 112

The r term is 3r.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3r + 2.25 + r2 = 112 + 2.25

Reorder the terms:
2.25 + 3r + r2 = 112 + 2.25

Combine like terms: 112 + 2.25 = 114.25
2.25 + 3r + r2 = 114.25

Factor a perfect square on the left side:
(r + 1.5)(r + 1.5) = 114.25

Calculate the square root of the right side: 10.688779163

Break this problem into two subproblems by setting 
(r + 1.5) equal to 10.688779163 and -10.688779163.

Subproblem 1

r + 1.5 = 10.688779163 Simplifying r + 1.5 = 10.688779163 Reorder the terms: 1.5 + r = 10.688779163 Solving 1.5 + r = 10.688779163 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = 10.688779163 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 10.688779163 + -1.5 r = 10.688779163 + -1.5 Combine like terms: 10.688779163 + -1.5 = 9.188779163 r = 9.188779163 Simplifying r = 9.188779163

Subproblem 2

r + 1.5 = -10.688779163 Simplifying r + 1.5 = -10.688779163 Reorder the terms: 1.5 + r = -10.688779163 Solving 1.5 + r = -10.688779163 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = -10.688779163 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -10.688779163 + -1.5 r = -10.688779163 + -1.5 Combine like terms: -10.688779163 + -1.5 = -12.188779163 r = -12.188779163 Simplifying r = -12.188779163

Solution

The solution to the problem is based on the solutions from the subproblems. r = {9.188779163, -12.188779163}

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