SOLUTION: please help me with this polynomial. z^2+9z+20 thank you and please show work so i know how to complete it on my own.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: please help me with this polynomial. z^2+9z+20 thank you and please show work so i know how to complete it on my own.      Log On


   



Question 186066: please help me with this polynomial.

z^2+9z+20

thank you and please show work so i know how to complete it on my own.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
This can be done by the method
known as
-completing the square-
Take 1%2F2 of the coefficient of
the z term, square it, then
add the results to both sides
-------------------------------
z%5E2+%2B+9z+%2B+20+=+0
But first, I have to get the constant
term on the other side. Subtract 20
from both sides
z%5E2+%2B+9z+=+-20
Now complete the square
z%5E2+%2B+9z+%2B+%289%2F2%29%5E2+=+-20+%2B+%289%2F2%29%5E2
z%5E2+%2B+9z+%2B+81%2F4+=+-20+%2B+81%2F4
With this method, the left side always ends up
being a perfect square. In this problem, both
sides are a perfect square
%28z+%2B+%289%2F2%29%29%5E2+=+-%2880%2F4%29+%2B+81%2F4
%28z+%2B+%289%2F2%29%29%5E2+=+1%2F4
Now take the square root of both sides,
both the (+) and (-) square root
z+%2B+9%2F2+=+1%2F2
z+=+-%289%2F2%29+%2B+1%2F2
z+=+-4
and
z+%2B+9%2F2+=+-%281%2F2%29
z+=+-%289%2F2%29+-%281%2F2%29
z+=+-5
-4 and -5 are the roots of the polynomial
To factor z%5E2+%2B+9z+%2B+20,
z+=+-4
z+-%28-4%29+=+0
z+%2B+4+=+0
and
z+-%28-5%29+=+0
z+%2B+5+=+0
The factoring is:
%28z+%2B+4%29%28z+%2B+5%29+=+z%5E2+%2B+9z+%2B+20