r^2+3r+0.5=0

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Solution for r^2+3r+0.5=0 equation:


Simplifying
r2 + 3r + 0.5 = 0

Reorder the terms:
0.5 + 3r + r2 = 0

Solving
0.5 + 3r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 3r + -0.5 + r2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 3r + r2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 3r + r2 = 0 + -0.5
3r + r2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
3r + r2 = -0.5

The r term is 3r.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3r + 2.25 + r2 = -0.5 + 2.25

Reorder the terms:
2.25 + 3r + r2 = -0.5 + 2.25

Combine like terms: -0.5 + 2.25 = 1.75
2.25 + 3r + r2 = 1.75

Factor a perfect square on the left side:
(r + 1.5)(r + 1.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(r + 1.5) equal to 1.322875656 and -1.322875656.

Subproblem 1

r + 1.5 = 1.322875656 Simplifying r + 1.5 = 1.322875656 Reorder the terms: 1.5 + r = 1.322875656 Solving 1.5 + r = 1.322875656 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = 1.322875656 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 1.322875656 + -1.5 r = 1.322875656 + -1.5 Combine like terms: 1.322875656 + -1.5 = -0.177124344 r = -0.177124344 Simplifying r = -0.177124344

Subproblem 2

r + 1.5 = -1.322875656 Simplifying r + 1.5 = -1.322875656 Reorder the terms: 1.5 + r = -1.322875656 Solving 1.5 + r = -1.322875656 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = -1.322875656 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -1.322875656 + -1.5 r = -1.322875656 + -1.5 Combine like terms: -1.322875656 + -1.5 = -2.822875656 r = -2.822875656 Simplifying r = -2.822875656

Solution

The solution to the problem is based on the solutions from the subproblems. r = {-0.177124344, -2.822875656}

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