Question

Part A: The area of a square is (9x^2 + 24x + 16) square units. Determine the length of each side of the square by factoring the area expression completely. Show your work. (5 points)

Part B: The area of a rectangle is (16x^2 − 25y^2) square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work. (5 points)

Answer 1

@Elsa213 @EvieSwan2405 @JustSaiyan @563blackghost

Answer 2

@Shadow

Answer 3

Do you know how to factor?

Answer 4

yes i believe so

Answer 5

@Shadow

Answer 6

Since the area of a square and rectangle is determine by multiply their length by their width, if you factor both the expressions completely, we'll have the lengths and heights of the sides.

Answer 7

Do you know what FOILing is?

Answer 8

no

Answer 9

First
Outer
Inner
Last

Basically it means this:
F - You multiply the first term in each of the parentheses by each other
O - You multiply the outer term in each of the parentheses by each other
I - You multiply the inner term in each of the parentheses by each other
L - You multiply the last term in each of the parentheses by each other

Example:

(5x + 2)(x + 9)

F: 5x times x = 5x^2
O: 5x times 9 = 45x
I: 2 times x = 2x
L: 2 times 9 = 18

Add them all up, and my expression is:

\[5x^2 + 47x + 18\]

Answer 10

Does that make sense?

Answer 11

oh ok get it so lets do this one first:
The area of a square is (9x^2 + 24x + 16) square units. Determine the length of each side of the square by factoring the area expression completely. Show your work.

Answer 12

Do you know where to start?

Answer 13

not really, please guide me

Answer 14

if you want solve that one^^ and then show me the steps and everything and then ill try to follow your steps and do the other one :)

Answer 15

Mhm, okay

Answer 16

\[9x^2 + 24x + 16\]

We basically need to reverse FOIL, and that starts at the beginning. We know that the first term is created by multiply two numbers together, therefore we must list the factors of 9

Factors: 1, 3, 9
Where, 1 times 9 = 9
And 3 times 3 = 9

So we know that the first terms of our factored polynomial are either

\[(x + ?)(9x + ?)\]

or

\[(3x + ?)(3x + ?)\]

In my experience its best to start with the even ones first.

Next we have outer, and inner. We work with these together because the first terms of each polynomial multiply by these guys.

\[(3x + ?)(3x + ?)\]

Outer: 3x times ?
Inner: 3x times ?

After these numbers multiply, they must be able to add up to 24x.

Also, these last terms of each parentheses must be able to multiply, and equal 16
I notice that. It's usually easier to deal with the last term first, to determine what to put in the parentheses (when there are coefficients in front of the x in the first terms.

Factors: 1, 2, 4, 8, 16
1 times 16
2 times 8
4 times 4

After laying these out, look at Outer and Inner again. If 3 were to multiply these numbers, would they add up to 24.

16 makes the number to large (51x to be exact, 3x times 16 = 48x, 3x times 1 = 3x)
I want some more even numbers, so I look at 4.

3 times 4 = 12, a factor of 24 is 12. Where 2 x 12 = 24. So I know that my ? are 4.

\[(3x + 4)(3x + 4)\]

Answer 17

This process becomes easier and faster as you become more experienced at math. Like I didn't need to go through this process of listing factors as I recognized that it was 4 at the start. This just comes through having done it many times in different concepts, and your multiplication tables.

Answer 18

This is why people who cheat receive trouble later on, as they either can't do the concepts, or can only replicate what they've seen, not knowing the concept entirely. Lack of practice can also cost time on college entrance exams (SAT, ACT, etc).

Answer 19

ok i understand now i was practicing im back i think i got the answer for part b @Shadow

Answer 20

Lets hear it

Answer 21

16x^2 - 25y^2
= (4x + 5y )( 4x - 5y)
The dimensions of the rectangle is 4x + 5y and width = 4x - 5y

Answer 22

Correct