4r^2+12r+1=0

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Solution for 4r^2+12r+1=0 equation:


Simplifying
4r2 + 12r + 1 = 0

Reorder the terms:
1 + 12r + 4r2 = 0

Solving
1 + 12r + 4r2 = 0

Solving for variable 'r'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
0.25 + 3r + r2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 3r + -0.25 + r2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 3r + r2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 3r + r2 = 0 + -0.25
3r + r2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
3r + r2 = -0.25

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 = -0.25 + 2.25

Reorder the terms:
2.25 + 3r + r2 = -0.25 + 2.25

Combine like terms: -0.25 + 2.25 = 2
2.25 + 3r + r2 = 2

Factor a perfect square on the left side:
(r + 1.5)(r + 1.5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(r + 1.5) equal to 1.414213562 and -1.414213562.

Subproblem 1

r + 1.5 = 1.414213562 Simplifying r + 1.5 = 1.414213562 Reorder the terms: 1.5 + r = 1.414213562 Solving 1.5 + r = 1.414213562 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 = 1.414213562 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 1.414213562 + -1.5 r = 1.414213562 + -1.5 Combine like terms: 1.414213562 + -1.5 = -0.085786438 r = -0.085786438 Simplifying r = -0.085786438

Subproblem 2

r + 1.5 = -1.414213562 Simplifying r + 1.5 = -1.414213562 Reorder the terms: 1.5 + r = -1.414213562 Solving 1.5 + r = -1.414213562 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 = -1.414213562 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -1.414213562 + -1.5 r = -1.414213562 + -1.5 Combine like terms: -1.414213562 + -1.5 = -2.914213562 r = -2.914213562 Simplifying r = -2.914213562

Solution

The solution to the problem is based on the solutions from the subproblems. r = {-0.085786438, -2.914213562}

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