Factored form of the quadratic function

🏆Practice representation as a product

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 XX axis.
The factored form of the quadratic function looks like this:
y=(x−t)×(x−k)y=(x-t) \times (x-k)

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Find the standard representation of the following function

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

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Where
tt and kk are the intersection points of the parabola with the XX axis.
In the following way: (t,0)(k,0)(t,0) (k,0)
Let's see an example of the factored form:

y=(x−7)×(x+2)y=(x-7) \times (x+2)
We can determine that:

the intersection points with the XX axis are:
(7,0) (7,0) 
 (−2,0) (- 2,0)
Notice that, since there is a minus sign in the original form before kk and tt, we can deduce that if there is a plus sign before one of them it is negative and, therefore −2-2 and not 22.


Examples and exercises with solutions of the factored form of the quadratic function

Exercise #1

Determine the points of intersection of the function

y=(x−5)(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

Find the standard representation of the following function

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

Video Solution

Answer

f(x)=3x2+12x f(x)=3x^2+12x

Exercise #3

Find the standard representation of the following function

f(x)=(x+2)(x−4) f(x)=(x+2)(x-4)

Video Solution

Answer

f(x)=x2−2x−8 f(x)=x^2-2x-8

Exercise #4

Find the standard representation of the following function

f(x)=(x−2)(x+5) f(x)=(x-2)(x+5)

Video Solution

Answer

f(x)+x2+3x−10 f(x)+x^2+3x-10

Exercise #5

Find the standard representation of the following function

f(x)=(x−6)(x−2) f(x)=(x-6)(x-2)

Video Solution

Answer

f(x)=x2−8x+12 f(x)=x^2-8x+12

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