z^2-z-3=0

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


Simplifying
z2 + -1z + -3 = 0

Reorder the terms:
-3 + -1z + z2 = 0

Solving
-3 + -1z + 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 + -1z + 3 + z2 = 0 + 3

Reorder the terms:
-3 + 3 + -1z + z2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1z + z2 = 0 + 3
-1z + z2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1z + z2 = 3

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 3 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
0.25 + -1z + z2 = 3.25

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(z + -0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

z + -0.5 = 1.802775638 Simplifying z + -0.5 = 1.802775638 Reorder the terms: -0.5 + z = 1.802775638 Solving -0.5 + z = 1.802775638 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 1.802775638 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 1.802775638 + 0.5 z = 1.802775638 + 0.5 Combine like terms: 1.802775638 + 0.5 = 2.302775638 z = 2.302775638 Simplifying z = 2.302775638

Subproblem 2

z + -0.5 = -1.802775638 Simplifying z + -0.5 = -1.802775638 Reorder the terms: -0.5 + z = -1.802775638 Solving -0.5 + z = -1.802775638 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -1.802775638 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -1.802775638 + 0.5 z = -1.802775638 + 0.5 Combine like terms: -1.802775638 + 0.5 = -1.302775638 z = -1.302775638 Simplifying z = -1.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.302775638, -1.302775638}

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