what is relationship between the zeros of the quadratic function and the linear factors of the expression?

Question

I don't understand the relation ship between the two.it would be a big help so I can make a short summary of the relationship.

Answer 1

Let's use an example to help you understand.

For the quadratic function f(x) = x² + x - 30, the graph looks like this:

It can be factored as follows: f(x) = (x + 6)(x - 5).

In this factored form, the linear factors are "x + 6" and "x - 5"

The "zeros" of the quadratic equation occur when f(x) (or "y") equals zero. The place on the graph where y = 0 is the x-axis. So the zeros are the same as the x-intercepts.

When we make f(x) equal to zero, like this:

f(x) = (x + 6)(x - 5)

0 = (x + 6)(x - 5)

we can see that we are multiplying two linear factors together and the result is zero. The only way you can multiply two things together and get zero as a result is if one of the two things is equal to zero.

So if 0 = (x + 6)(x - 5) then (x + 6) = 0 or (x - 5) = 0.

If x + 6 = 0, then x = -6

If x - 5 = 0, then x = 5

We can see on the graph that these are the x-intercepts.

Answer 2

Hey Nonna,

This is a good question! The zeros of the quadratic function are the points where the functions outputs are zero. How do you usually solve for zeros of a function? You set the function equal to zero and solve it by factoring. For example: if you had:

f(x) = x²+2x-3

your zeros would satisfy the equation:

0 = x²+2x-3.

This can be solved by factoring:

0 = (x+3)(x-1)

which means this equation is true when x= -3 and x=1. Try to think what that means about the general relationship between the two.

So the relationship between the linear factors and the zeros of the function are that the zeros occur at the x's which make the linear factors zero. Hope this offers a bit of intuition about the problem.

-Nathan