1+4z+z^2=0

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Solution for 1+4z+z^2=0 equation:


Simplifying
1 + 4z + z2 = 0

Solving
1 + 4z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 4z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 4z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 4z + z2 = 0 + -1
4z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
4z + z2 = -1

The z term is 4z.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4z + 4 + z2 = -1 + 4

Reorder the terms:
4 + 4z + z2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + 4z + z2 = 3

Factor a perfect square on the left side:
(z + 2)(z + 2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(z + 2) equal to 1.732050808 and -1.732050808.

Subproblem 1

z + 2 = 1.732050808 Simplifying z + 2 = 1.732050808 Reorder the terms: 2 + z = 1.732050808 Solving 2 + z = 1.732050808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 1.732050808 + -2 Combine like terms: 2 + -2 = 0 0 + z = 1.732050808 + -2 z = 1.732050808 + -2 Combine like terms: 1.732050808 + -2 = -0.267949192 z = -0.267949192 Simplifying z = -0.267949192

Subproblem 2

z + 2 = -1.732050808 Simplifying z + 2 = -1.732050808 Reorder the terms: 2 + z = -1.732050808 Solving 2 + z = -1.732050808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -1.732050808 + -2 Combine like terms: 2 + -2 = 0 0 + z = -1.732050808 + -2 z = -1.732050808 + -2 Combine like terms: -1.732050808 + -2 = -3.732050808 z = -3.732050808 Simplifying z = -3.732050808

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.267949192, -3.732050808}

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