Finding zero points - Examples, Exercises and Solutions

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 Finding zero points

Exercise #1

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

To find the point of intersection with the X-axis, we will want 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 will check the possibilities.

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 #2

Determine the points of intersection of the function

y=x(x1) y=x(-x-1)

With the X

Video Solution

Answer

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

Exercise #3

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)

Exercise #4

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 #5

Consider the following function:

y=x(x1) y=x(x-1)

Determine the points of intersection with x.

Video Solution

Answer

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

Exercise #1

Determine the points of intersection of the function

y=(x1)(x+10) y=(x-1)(x+10)

With the X

Video Solution

Answer

(1,0),(10,0) (1,0),(-10,0)

Exercise #2

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 #3

Determine the points of intersection of the function

y=(x1)(x1) y=(x-1)(x-1)

With the X

Video Solution

Answer

(1,0) (1,0)

Exercise #4

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 #5

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 #1

Determine the points of intersection of the function

y=(x2)(x+3) y=(x-2)(x+3)

With the X

Video Solution

Answer

(3,0),(2,0) (-3,0),(2,0)

Exercise #2

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 #3

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 #4

Determine the points of intersection of the function

y=(x6)(x5) y=(x-6)(x-5)

With the X

Video Solution

Answer

(6,0),(5,0) (6,0),(5,0)

Exercise #5

Determine the points of intersection of the function

y=x(x+1) y=x(x+1)

With the X

Video Solution

Answer

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

Topics learned in later sections

  1. Vertex of a parabola
  2. Symmetry in a parabola