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 XX-axis.
To find them, we set Y=0 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 XX-axis at two different points.
  2. One result -
    In this case, the function intersects the XX-axis at only one point, meaning the vertex of the parabola is exactly on the XX-axis.
  3. No results -
    In this case, the function does not intersect the XX-axis at all, meaning it hovers above or below it.

Suggested Topics to Practice in Advance

  1. The quadratic function
  2. Parabola
  3. Plotting the Quadratic Function Using Parameters a, b and c

Practice Zeros of a Fuction

Examples with solutions for Zeros of a Fuction

Exercise #1

The following function has been graphed below:

f(x)=x28x+16 f(x)=x^2-8x+16

Calculate point C.

CCC

Video Solution

Step-by-Step Solution

To solve the exercise, first note that point C lies on the X-axis.

Therefore, to find it, we need to understand what is the X value when Y equals 0.

 

Let's set the equation equal to 0:

0=x²-8x+16

We'll use the preferred method (trinomial or quadratic formula) to find the X values, and we'll discover that

X=4

 

Answer

(4,0) (4,0)

Exercise #2

Determine the points of intersection of the function

y=(x5)(x+5) y=(x-5)(x+5)

With the X

Video Solution

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!

Answer

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

Exercise #3

The following function has been graphed below:

f(x)=x2+5x+6 f(x)=-x^2+5x+6

Calculate points A and B.

BBBAAACCC

Video Solution

Answer

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

Exercise #4

The following function has been graphed below:

f(x)=x26x+5 f(x)=x^2-6x+5

Calculate points A and B.

AAABBB

Video Solution

Answer

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

Exercise #5

The following function has been graphed below:

f(x)=x28x+16 f(x)=x^2-8x+16

Calculate point A.

AAACCC

Video Solution

Answer

(0,16) (0,16)

Exercise #6

The following function has been graphed below:

f(x)=x23x4 f(x)=x^2-3x-4

Calculate points A and B.

CCCAAABBB

Video Solution

Answer

A(1,0),B(4,0) A(-1,0),B(4,0)

Exercise #7

The following function has been graphed below:

f(x)=x26x f(x)=x^2-6x

Calculate points A and B.

AAABBB

Video Solution

Answer

(6,0),(0,0) (6,0),\lparen0,0)

Exercise #8

The following function has been graphed below:

f(x)=x26x+8 f(x)=x^2-6x+8

Calculate points A and B.

AAABBB

Video Solution

Answer

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

Exercise #9

Determine the points of intersection of the function

y=(x3)(x+3) y=(x-3)(x+3)

With the X

Video Solution

Answer

(3,0),(3,0) (-3,0),(3,0)

Exercise #10

Determine the points of intersection of the function

y=x(x+5) y=x(x+5)

With the X

Video Solution

Answer

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

Exercise #11

Determine the points of intersection of the function

y=(x+7)(x+2) y=(x+7)(x+2)

With the X

Video Solution

Answer

(2,0),(7,0) (-2,0),(-7,0)

Exercise #12

Determine the points of intersection of the function

y=(x+3)(x3) y=(x+3)(x-3)

With the X

Video Solution

Answer

(3,0),(3,0) (3,0),(-3,0)

Exercise #13

Determine the points of intersection of the function

y=(x11)(x+1) y=(x-11)(x+1)

With the X

Video Solution

Answer

(1,0),(11,0) (-1,0),(11,0)

Exercise #14

Determine the points of intersection of the function

y=(x+8)(x9) y=(x+8)(x-9)

With the X

Video Solution

Answer

(8,0),(9,0) (-8,0),(9,0)

Exercise #15

Determine the points of intersection of the function

y=(4x+8)(x+1) y=(4x+8)(x+1)

With the X

Video Solution

Answer

(1,0),(2,0) (-1,0),(-2,0)