# Finding the Zeros of a Parabola

🏆Practice zeros of a fuction

## Finding the zeros of a quadratic function of the form $$y=ax^2+bx+c$$

Zero points of a function are its intersection points with the $X$-axis.
To find them, we set $Y=0$,
we get an equation that can sometimes be solved using a trinomial or the quadratic formula.

### When trying to find the zero point, you can encounter three possible results:

1. Two results -
In this case, the function intersects the $X$-axis at two different points.
2. One result -
In this case, the function intersects the $X$-axis at only one point, meaning the vertex of the parabola is exactly on the $X$-axis.
In this case, the function does not intersect the $X$-axis at all, meaning it hovers above or below it.

## Test yourself on zeros of a fuction!

The following function has been graphed below:

$$f(x)=-x^2+5x+6$$

Calculate points A and B.

Zero points describe a situation where the function equals zero.

Let's look at an example:
We have the function
$X^2-4x+5$
We substitute into the quadratic formula and get:

${-4 \pm \sqrt{(-4)^2-4*1*5} \over 21}=-\frac{4\pm\sqrt{-4}}{2}$

This equation has no solution because the delta inside the root is negative. Therefore, this equation never equals zero, it hovers above the X-axis, and it has no zero points.

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## Examples with solutions for Zeros of a Fuction

### Exercise #1

Determine the points of intersection of the function

$y=(x-5)(x+5)$

With the X

### Step-by-Step Solution

In order to find the point of the intersection with the X-axis, we first need to establish that Y=0.

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

When we have an equation of this type, we know that one of these parentheses must be equal to 0, so we begin by checking the possible options.

x-5 = 0
x = 5

x+5 = 0
x = -5

That is, we have two points of intersection with the x-axis, when we discover their x points, and the y point is already known to us (0, as we placed it):

(5,0)(-5,0)

This is the solution!

$(5,0),(-5,0)$

### Exercise #2

The following function has been graphed below:

$f(x)=-x^2+5x+6$

Calculate points A and B.

### Video Solution

$(-1,0),(6,0)$

### Exercise #3

The following function has been graphed below:

$f(x)=x^2-6x+8$

Calculate points A and B.

### Video Solution

$(2,0),(4,0)$

### Exercise #4

The following function has been graphed below:

$f(x)=x^2-8x+16$

Calculate point A.

### Video Solution

$(0,16)$

### Exercise #5

The following function has been graphed below:

$f(x)=x^2-6x+5$

Calculate points A and B.



### Video Solution

$(1,0),(5,0)$