z^2-4z-3=0

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Solution for z^2-4z-3=0 equation:


Simplifying
z2 + -4z + -3 = 0

Reorder the terms:
-3 + -4z + z2 = 0

Solving
-3 + -4z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -4z + 3 + z2 = 0 + 3

Reorder the terms:
-3 + 3 + -4z + z2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -4z + z2 = 0 + 3
-4z + z2 = 0 + 3

Combine like terms: 0 + 3 = 3
-4z + z2 = 3

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 = 3 + 4

Reorder the terms:
4 + -4z + z2 = 3 + 4

Combine like terms: 3 + 4 = 7
4 + -4z + z2 = 7

Factor a perfect square on the left side:
(z + -2)(z + -2) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(z + -2) equal to 2.645751311 and -2.645751311.

Subproblem 1

z + -2 = 2.645751311 Simplifying z + -2 = 2.645751311 Reorder the terms: -2 + z = 2.645751311 Solving -2 + z = 2.645751311 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 = 2.645751311 + 2 Combine like terms: -2 + 2 = 0 0 + z = 2.645751311 + 2 z = 2.645751311 + 2 Combine like terms: 2.645751311 + 2 = 4.645751311 z = 4.645751311 Simplifying z = 4.645751311

Subproblem 2

z + -2 = -2.645751311 Simplifying z + -2 = -2.645751311 Reorder the terms: -2 + z = -2.645751311 Solving -2 + z = -2.645751311 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 = -2.645751311 + 2 Combine like terms: -2 + 2 = 0 0 + z = -2.645751311 + 2 z = -2.645751311 + 2 Combine like terms: -2.645751311 + 2 = -0.645751311 z = -0.645751311 Simplifying z = -0.645751311

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.645751311, -0.645751311}

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