r^2-6r+7=0

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Solution for r^2-6r+7=0 equation:


Simplifying
r2 + -6r + 7 = 0

Reorder the terms:
7 + -6r + r2 = 0

Solving
7 + -6r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -6r + -7 + r2 = 0 + -7

Reorder the terms:
7 + -7 + -6r + r2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -6r + r2 = 0 + -7
-6r + r2 = 0 + -7

Combine like terms: 0 + -7 = -7
-6r + r2 = -7

The r term is -6r.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6r + 9 + r2 = -7 + 9

Reorder the terms:
9 + -6r + r2 = -7 + 9

Combine like terms: -7 + 9 = 2
9 + -6r + r2 = 2

Factor a perfect square on the left side:
(r + -3)(r + -3) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(r + -3) equal to 1.414213562 and -1.414213562.

Subproblem 1

r + -3 = 1.414213562 Simplifying r + -3 = 1.414213562 Reorder the terms: -3 + r = 1.414213562 Solving -3 + r = 1.414213562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + r = 1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + r = 1.414213562 + 3 r = 1.414213562 + 3 Combine like terms: 1.414213562 + 3 = 4.414213562 r = 4.414213562 Simplifying r = 4.414213562

Subproblem 2

r + -3 = -1.414213562 Simplifying r + -3 = -1.414213562 Reorder the terms: -3 + r = -1.414213562 Solving -3 + r = -1.414213562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + r = -1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + r = -1.414213562 + 3 r = -1.414213562 + 3 Combine like terms: -1.414213562 + 3 = 1.585786438 r = 1.585786438 Simplifying r = 1.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.414213562, 1.585786438}

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