4x^2+6x+1=0

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Solution for 4x^2+6x+1=0 equation:


Simplifying
4x2 + 6x + 1 = 0

Reorder the terms:
1 + 6x + 4x2 = 0

Solving
1 + 6x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
0.25 + 1.5x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 1.5x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 1.5x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 1.5x + x2 = 0 + -0.25
1.5x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
1.5x + x2 = -0.25

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = -0.25 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = -0.25 + 0.5625

Combine like terms: -0.25 + 0.5625 = 0.3125
0.5625 + 1.5x + x2 = 0.3125

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(x + 0.75) equal to 0.559016994 and -0.559016994.

Subproblem 1

x + 0.75 = 0.559016994 Simplifying x + 0.75 = 0.559016994 Reorder the terms: 0.75 + x = 0.559016994 Solving 0.75 + x = 0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 0.559016994 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 0.559016994 + -0.75 x = 0.559016994 + -0.75 Combine like terms: 0.559016994 + -0.75 = -0.190983006 x = -0.190983006 Simplifying x = -0.190983006

Subproblem 2

x + 0.75 = -0.559016994 Simplifying x + 0.75 = -0.559016994 Reorder the terms: 0.75 + x = -0.559016994 Solving 0.75 + x = -0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -0.559016994 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -0.559016994 + -0.75 x = -0.559016994 + -0.75 Combine like terms: -0.559016994 + -0.75 = -1.309016994 x = -1.309016994 Simplifying x = -1.309016994

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.190983006, -1.309016994}

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