r^2-10r+3=0

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


Simplifying
r2 + -10r + 3 = 0

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

Solving
3 + -10r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -10r + -3 + r2 = 0 + -3

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

Combine like terms: 3 + -3 = 0
0 + -10r + r2 = 0 + -3
-10r + r2 = 0 + -3

Combine like terms: 0 + -3 = -3
-10r + r2 = -3

The r term is -10r.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10r + 25 + r2 = -3 + 25

Reorder the terms:
25 + -10r + r2 = -3 + 25

Combine like terms: -3 + 25 = 22
25 + -10r + r2 = 22

Factor a perfect square on the left side:
(r + -5)(r + -5) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(r + -5) equal to 4.69041576 and -4.69041576.

Subproblem 1

r + -5 = 4.69041576 Simplifying r + -5 = 4.69041576 Reorder the terms: -5 + r = 4.69041576 Solving -5 + r = 4.69041576 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + r = 4.69041576 + 5 Combine like terms: -5 + 5 = 0 0 + r = 4.69041576 + 5 r = 4.69041576 + 5 Combine like terms: 4.69041576 + 5 = 9.69041576 r = 9.69041576 Simplifying r = 9.69041576

Subproblem 2

r + -5 = -4.69041576 Simplifying r + -5 = -4.69041576 Reorder the terms: -5 + r = -4.69041576 Solving -5 + r = -4.69041576 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + r = -4.69041576 + 5 Combine like terms: -5 + 5 = 0 0 + r = -4.69041576 + 5 r = -4.69041576 + 5 Combine like terms: -4.69041576 + 5 = 0.30958424 r = 0.30958424 Simplifying r = 0.30958424

Solution

The solution to the problem is based on the solutions from the subproblems. r = {9.69041576, 0.30958424}

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