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

Answer 1

Hello, Nonna

Factoring a quadratic is one of the best methods for finding its zeros.

If you set a quadratic equal to zero, then the values of x which satisfy that equation will be your zeros. For example, take the following quadratic:

x² + 3x +2

1. Set it equal to zero: x² + 3x +2 = 0

Now if we factor the expression on the left-hand side we can find the x values that satisfy the relationship.

(x + 1)(x +2) = 0, meaning that x = -1, and x = -2 are both solutions and thus, the function's zeros.

The important part, however, is to remember that these zeros and the linear factors that we got them from have the opposite sign. This makes sense of course, because the value of x plus the second term in its factor must equal zero.

(Note, some solutions may not be real numbers but imaginary, meaning that they don't intersect the x-axis. In order to be a real zero of a function, x must be a real number.)

I hope this helps!

Daniel

Answer 2

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.