Product Representation - Examples, Exercises and Solutions

Factored form of the quadratic function

This form is called factored because it uses the factors of a multiplication.

With this form, we can easily identify the points of intersection of the function with the $X$ axis.
The factored form of the quadratic function looks like this:
$y=(x-t) \times (x-k)$

Suggested Topics to Practice in Advance

1. Standard Form of the Quadratic Function

Examples with solutions for Product Representation

Exercise #1

Find the standard representation of the following function

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

Step-by-Step Solution

We will use the expanded distributive property.

(a+1)⋆(b+2) = ab+2a+b+2

Let's substitute the given values and solve:

(x-2)(x+5) =

x²-2x+5x+-2*5=

x²+3x-10

And that's the solution!

$f(x)+x^2+3x-10$

Exercise #2

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 #3

Find the standard representation of the following function

$f(x)=3x(x+4)$

Video Solution

$f(x)=3x^2+12x$

Exercise #4

Find the standard representation of the following function

$f(x)=(x+2)(x-4)$

Video Solution

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

Exercise #5

Find the standard representation of the following function

$f(x)=(x-6)(x-2)$

Video Solution

$f(x)=x^2-8x+12$

Exercise #6

Find the standard representation of the following function

$f(x)=-x(x-8)$

Video Solution

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

Exercise #7

Does the parable

$y=(x-2)(x+1)$

Is there a minimum or maximum point?

Minimal point

Exercise #8

Determine the points of intersection of the function

$y=x(-x-1)$

With the X

Video Solution

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

Exercise #9

Determine the points of intersection of the function

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

With the X

Video Solution

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

Exercise #10

Determine the points of intersection of the function

$y=x(x+5)$

With the X

Video Solution

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

Exercise #11

Consider the following function:

$y=x(x-1)$

Determine the points of intersection with x.

Video Solution

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

Exercise #12

Determine the points of intersection of the function

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

With the X

Video Solution

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

Exercise #13

Determine the points of intersection of the function

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

With the X

Video Solution

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

Exercise #14

Determine the points of intersection of the function

$y=(x-1)(x-1)$

With the X

Video Solution

$(1,0)$

Exercise #15

Determine the points of intersection of the function

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

With the X

Video Solution

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