r^2-7r=-3

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


Simplifying
r2 + -7r = -3

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

Solving
-7r + r2 = -3

Solving for variable 'r'.

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

Combine like terms: -3 + 3 = 0
3 + -7r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -7r + -3 + r2 = 0 + -3

Reorder the terms:
3 + -3 + -7r + r2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -7r + r2 = 0 + -3
-7r + r2 = 0 + -3

Combine like terms: 0 + -3 = -3
-7r + r2 = -3

The r term is -7r.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7r + 12.25 + r2 = -3 + 12.25

Reorder the terms:
12.25 + -7r + r2 = -3 + 12.25

Combine like terms: -3 + 12.25 = 9.25
12.25 + -7r + r2 = 9.25

Factor a perfect square on the left side:
(r + -3.5)(r + -3.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(r + -3.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

r + -3.5 = 3.041381265 Simplifying r + -3.5 = 3.041381265 Reorder the terms: -3.5 + r = 3.041381265 Solving -3.5 + r = 3.041381265 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + r = 3.041381265 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + r = 3.041381265 + 3.5 r = 3.041381265 + 3.5 Combine like terms: 3.041381265 + 3.5 = 6.541381265 r = 6.541381265 Simplifying r = 6.541381265

Subproblem 2

r + -3.5 = -3.041381265 Simplifying r + -3.5 = -3.041381265 Reorder the terms: -3.5 + r = -3.041381265 Solving -3.5 + r = -3.041381265 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + r = -3.041381265 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + r = -3.041381265 + 3.5 r = -3.041381265 + 3.5 Combine like terms: -3.041381265 + 3.5 = 0.458618735 r = 0.458618735 Simplifying r = 0.458618735

Solution

The solution to the problem is based on the solutions from the subproblems. r = {6.541381265, 0.458618735}

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