3r^2+10r-96=0

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Solution for 3r^2+10r-96=0 equation:


Simplifying
3r2 + 10r + -96 = 0

Reorder the terms:
-96 + 10r + 3r2 = 0

Solving
-96 + 10r + 3r2 = 0

Solving for variable 'r'.

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-32 + 3.333333333r + r2 = 0

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + 3.333333333r + 32 + r2 = 0 + 32

Reorder the terms:
-32 + 32 + 3.333333333r + r2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + 3.333333333r + r2 = 0 + 32
3.333333333r + r2 = 0 + 32

Combine like terms: 0 + 32 = 32
3.333333333r + r2 = 32

The r term is 3.333333333r.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333r + 2.777777779 + r2 = 32 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333r + r2 = 32 + 2.777777779

Combine like terms: 32 + 2.777777779 = 34.777777779
2.777777779 + 3.333333333r + r2 = 34.777777779

Factor a perfect square on the left side:
(r + 1.666666667)(r + 1.666666667) = 34.777777779

Calculate the square root of the right side: 5.897268671

Break this problem into two subproblems by setting 
(r + 1.666666667) equal to 5.897268671 and -5.897268671.

Subproblem 1

r + 1.666666667 = 5.897268671 Simplifying r + 1.666666667 = 5.897268671 Reorder the terms: 1.666666667 + r = 5.897268671 Solving 1.666666667 + r = 5.897268671 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + r = 5.897268671 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + r = 5.897268671 + -1.666666667 r = 5.897268671 + -1.666666667 Combine like terms: 5.897268671 + -1.666666667 = 4.230602004 r = 4.230602004 Simplifying r = 4.230602004

Subproblem 2

r + 1.666666667 = -5.897268671 Simplifying r + 1.666666667 = -5.897268671 Reorder the terms: 1.666666667 + r = -5.897268671 Solving 1.666666667 + r = -5.897268671 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + r = -5.897268671 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + r = -5.897268671 + -1.666666667 r = -5.897268671 + -1.666666667 Combine like terms: -5.897268671 + -1.666666667 = -7.563935338 r = -7.563935338 Simplifying r = -7.563935338

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.230602004, -7.563935338}

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