Question 146643
hello && greetings please help !!

Which conic section is represented by the equation {{{4x^2-4xy+y^2+4x-2y+1=0}}} ????

A. ELLIPSE
B. 2 LINES
C. HYPERBOLA
D. 1 LINE
E. NONE OF THESE 
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{{{4x^2-4xy+y^2+4x-2y+1=0}}} 

Group the first three terms and
group the fourth and fifth terms:

{{{(4x^2-4xy+y^2)+(4x-2y)+1=0}}}

Factor the trinomial in the first parentheses

{{{(2x-y)(2x-y)+(4x-2y)+1=0}}}

Write {{{(2x-y)(2x-y)}}} as {{{(2x-y)^2}}}

{{{(2x-y)^2+(4x-2y)+1=0}}}

Factor 2 out of the second parentheses:

{{{(2x-y)^2+2(2x-y)+1=0}}}

Let {{{z=(2x-y)}}}

{{{z^2+2z+1=0}}}

Factor that as

{{{(z+1)(z+1)=0}}}

{{{(z+1)^2=0}}}

Use the square root principle:

{{{sqrt((z+1)^2)}}}=±{{{sqrt(0)}}}

{{{z+1=0}}}

Since {{{z=2x-y}}}, replace {{{z}}} by {{{2x-y}}}

{{{2x-y+1=0}}}

This the equation of one line:

So the correct choice is D

Edwin</pre>