What are the zeros of the parabola:

Question

What are the zeros of the parabola: f(x) = x² - 7x + 10 (that is, what are the x-intercepts or the points where the graph crosses the x-axis)

Answer 1

If a quadratic is factorable, finding the zeros is simple. There are several methods used to factor a quadratic and I'm unsure what method you may have been taught but can you see that x² - 7x + 10 = (x - 5)(x - 2)? Now, to find the zeros, set the quadratic equal to zero:

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

This means that either "x - 5" equals zero or that "x - 2" equals zero:

x - 5 = 0 means x = 5

x - 2 = 0 means x = 2

So the zeros are x = 5 and x = 2

Answer 2

Hi Yittel, I hope you are doing well.

First, let's break down: x^2 - 7x + 10

The first thing you will notice about this equation is that it is factorable which means that we can find the zeroes or otherwise known as the x-intercepts.

So how can we factor this polynomial out.

I like to use what is called the "X" method where you find two numbers that when multiplied equals ten and when added equals -7.

We know that the factored form will take the form of (x-a)(x-b)

So we need ab = 10 and a + b = -7.

This will only work with a = -2 and b = -5 (or if a = -5 and b = -2).

So now let's plug this into (x+a)(x+b)

We will soon get (x-2)(x-5)

We know that at the x-intercept, y is equal to zero.

Therefore, we will set (x-2)(x-5) equal to zero to find the x-intercept.

For this to equal to zero, one of two things must occur.

1) x - 2 = 0

x = 2

or

2) x - 5 = 0

x =5

There are two solutions to this equation because x can be equal to 2 or 5 to get a y-value of zero.

The x-intercepts are (5, 0) and (2,0).

Hope that helps!