How do you find all the zeros of $f(x) = x^2 - 12x + 20$ with its multiplicities?

Question

1520 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-27T19:01:36

$f(x)=0$ when $x=10$ or $x=2$ each with multiplicity $1$

Here are a couple of methods:

$f(x) = x^2-12x+20$

$=(x-6)^2-36+20$

$=(x-6)^2-16$

$=(x-6)^2-4^2$

$=((x-6)-4)((x-6)+4)$

$=(x-10)(x-2)$

So $f(x)=0$ when $x=10$ or $x=2$ with multiplicity $1$

$color(white)()$
Product and sum

Note that $10*2 = 20$ and $10+2 = 12$

So:

$f(x) = x^2-12+20 = x^2-(10+2)x+(10*2) = (x-10)(x-2)$

So $f(x)=0$ when $x=10$ or $x=2$ with multiplicity $1$

How do you find all the zeros of f(x) = x^2 - 12x + 20f(x) = x^2 - 12x + 20 with its multiplicities?