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 = -0.30958424 r = -0.30958424 Simplifying r = -0.30958424

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 = -9.69041576 r = -9.69041576 Simplifying r = -9.69041576

Solution

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

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