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:
$y=(x-t) \times (x-k)$

Detailed explanation

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Find the standard representation of the following function

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

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

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

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

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

Exercise #1

Find the standard representation of the following function

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

Video Solution

Step-by-Step Solution

We will begin by using the distributive property in order to expand the following expression.

(a+1)⋆(b+2) = ab+2a+b+2

We will then proceed to insert the known values into the equation and solve as follows:

(x-2)(x+5) =

x²-2x+5x+-2*5=

x²+3x-10

And that's the solution!

Answer

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

Exercise #2

Determine the points of intersection of the function

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

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!