Question 186066
This can be done by the method
known as 
-completing the square-
Take {{{1/2}}} of the coefficient of 
the {{{z}}} term, square it, then
add the results to both sides 
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{{{z^2 + 9z + 20 = 0}}}
But first, I have to get the constant 
term on the other side. Subtract {{{20}}}
from both sides
{{{z^2 + 9z = -20}}}
Now complete the square
{{{z^2 + 9z + (9/2)^2 = -20 + (9/2)^2}}}
{{{z^2 + 9z + 81/4 = -20 + 81/4}}}
With this method, the left side always ends up
being a perfect square. In this problem, both
sides are a perfect square
{{{(z + (9/2))^2 = -(80/4) + 81/4}}}
{{{(z + (9/2))^2 = 1/4}}}
Now take the square root of both sides,
both the (+) and (-) square root
{{{z + 9/2 = 1/2}}}
{{{z = -(9/2) + 1/2}}}
{{{z = -4}}}
and
{{{z + 9/2 = -(1/2)}}}
{{{z = -(9/2) -(1/2)}}}
{{{z = -5}}}
{{{-4}}} and {{{-5}}} are the roots of the polynomial
To factor {{{z^2 + 9z + 20}}},
{{{z = -4}}}
{{{z -(-4) = 0}}}
{{{z + 4 = 0}}}
and
{{{z -(-5) = 0}}}
{{{z + 5 = 0}}}
The factoring is:
{{{(z + 4)(z + 5) = z^2 + 9z + 20}}}