3r^2-13r-7=5

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Solution for 3r^2-13r-7=5 equation:


Simplifying
3r2 + -13r + -7 = 5

Reorder the terms:
-7 + -13r + 3r2 = 5

Solving
-7 + -13r + 3r2 = 5

Solving for variable 'r'.

Reorder the terms:
-7 + -5 + -13r + 3r2 = 5 + -5

Combine like terms: -7 + -5 = -12
-12 + -13r + 3r2 = 5 + -5

Combine like terms: 5 + -5 = 0
-12 + -13r + 3r2 = 0

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

Divide each side by '3'.
-4 + -4.333333333r + r2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -4.333333333r + 4 + r2 = 0 + 4

Reorder the terms:
-4 + 4 + -4.333333333r + r2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -4.333333333r + r2 = 0 + 4
-4.333333333r + r2 = 0 + 4

Combine like terms: 0 + 4 = 4
-4.333333333r + r2 = 4

The r term is -4.333333333r.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333r + 4.694444446 + r2 = 4 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333r + r2 = 4 + 4.694444446

Combine like terms: 4 + 4.694444446 = 8.694444446
4.694444446 + -4.333333333r + r2 = 8.694444446

Factor a perfect square on the left side:
(r + -2.166666667)(r + -2.166666667) = 8.694444446

Calculate the square root of the right side: 2.948634336

Break this problem into two subproblems by setting 
(r + -2.166666667) equal to 2.948634336 and -2.948634336.

Subproblem 1

r + -2.166666667 = 2.948634336 Simplifying r + -2.166666667 = 2.948634336 Reorder the terms: -2.166666667 + r = 2.948634336 Solving -2.166666667 + r = 2.948634336 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + r = 2.948634336 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + r = 2.948634336 + 2.166666667 r = 2.948634336 + 2.166666667 Combine like terms: 2.948634336 + 2.166666667 = 5.115301003 r = 5.115301003 Simplifying r = 5.115301003

Subproblem 2

r + -2.166666667 = -2.948634336 Simplifying r + -2.166666667 = -2.948634336 Reorder the terms: -2.166666667 + r = -2.948634336 Solving -2.166666667 + r = -2.948634336 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + r = -2.948634336 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + r = -2.948634336 + 2.166666667 r = -2.948634336 + 2.166666667 Combine like terms: -2.948634336 + 2.166666667 = -0.781967669 r = -0.781967669 Simplifying r = -0.781967669

Solution

The solution to the problem is based on the solutions from the subproblems. r = {5.115301003, -0.781967669}

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