Zero Product Property Problems - Please help

Question

I just want to know how to do these problems since my teacher didn't explain very well. She only gave the class the answers.

 

f(x)=x^2-7x+10

f(x)=4x^2+4x-24

f(x)=t^2+7t-60

Answer 1

Assuming you want to find the zeroes of each, you'll set them equal to 0, factor, and solve.

 

For the first one, you want factors of (+10) that add up to (-7):

x² - 7x + 10 = 0

 

Think for a while, write them out if you need to, and you'll come up with (-2)*(-5) = 10; (-2) + (-5) = -7

Written as factors, this becomes

(x-2)(x-5) = 0

 

If you have a product of two numbers that equal zero, what must be true about at least one of the factors? Consider this for different cases. Try multiplying different numbers together to get 0. The only possible way for this to be true is if at least one of the numbers to begin with is 0.

 

Therefore, since (x-2)(x-5) = 0, then either (x-2) must equal 0, or (x-5) must equal 0. If neither of them is 0, then the product will also be non-zero.

 

So, if x-2=0, then x=2, and if x-5=0, then x=5. These are your two zeroes to this function. You can do the same thing for the other two functions.