Question

When solving this equation, y = x² + 6x + 5, compare the method of factoring to the quadratic formula to find the roots. Which way seems easier and do you think it will always be this way?

Answer 1

lets see factoring methid..
can you think of two numbers a, b such trhat

a times b = 5
a plus b = 6

Answer 2

5x5=1? and 3+3=6?

Answer 3

5 times 5 = 25

Answer 4

sorry I meant 5x1

Answer 5

lets say the numbers are 1 and 5
then
1 times 5 = 5
and
1 plus 5 = 6

Answer 6

okay

Answer 7

and this satisfies our quadratic.
remember, you have to tick with two same numbers
\[y = x^2+1x+5x+5\\
y = x(x+1)+5(x+1)\\
y=(x+1)(x+5)
\]
so, we were easily able to factorize it

Answer 8

now, using the quadratic formula,
\[x={-b\pm\sqrt{b^2-4ac}\over2a}\\
a=1\quad b=6\quad c=5\\
x=\frac{-6\pm\sqrt{36-4(1)(5)}}{2(1)}\\
x=\frac{-6\pm\sqrt{16}}{2}\\
x={-6\pm4\over2}\\
x={-6+4\over2}\qquad x={-6-4\over2}\\
x=-1\qquad x=-5
\]

Answer 9

which one is easier?

Answer 10

I would say factoring

Answer 11

good. but can you always use that?

Answer 12

no?

Answer 13

good and why not?

Answer 14

because you may not always be able to find two numbers that satisfy the consitions
for example:
\[y=x^2+x+1\]
can you factorize this

Answer 15

no

Answer 16

\[y=x^2+2x+1\]can you factorize this?