Question

Write each trinomial as a perfect square?

Answer 1

x²+4x+4

Answer 2

x^2 + 4x + 4

x^2 + 2x + 2x + 4

(x^2 + 2x) + (2x + 4)

x(x+2) + 2(x+2)

(x+2)(x+2)

(x+2)^2


So x^2 + 4x + 4 written as a perfect square is (x+2)^2

Answer 3

Well, what about x²-16x+64?

Answer 4

x^2 - 16x + 64

x^2 - 8x - 8x + 64

(x^2-8x)+(-8x+64)

x(x-8)-8(x-8)

(x-8)(x-8)

(x-8)^2


So x^2 - 16x + 64 as a perfect square is (x-8)^2

Answer 5

Each time, I'm breaking up the x term into two equal pieces. Then I'm factoring by grouping.

Answer 6

I'm really slow at learning, and have to go through it like, step by step like i'm 5 years old.. So, if you don't want to do that I won't be offended :b I just can't follow those steps without them being explained because i'm so in the dark on this.

Answer 7

alright, I'll go back to the top and explain what I'm doing. My apologies.

Answer 8

Not your fault. I'm just not that smart, or quick at picking up stuff.

Answer 9

x^2 + 4x + 4 ... Start with the given expression


x^2 + 2x + 2x + 4 ... Break up 4x into 2x + 2x


(x^2+2x)+(2x+4) ... Group the terms into two groups


x(x+2)+(2x+4) ... Factor out x from the first group


x(x+2)+2(x+2) ... Factor out 2 from the second group


(x+2)(x+2) ... Factor out the GCF x+2


(x+2)^2 ... Condense the factors


So x^2 + 4x + 4 written as a perfect square is (x+2)^2

Answer 10

I'm sure you're doing fine, just keep practicing until you get it.

Answer 11

Okay, what do you mean by factor out the x? How do you do that?

Answer 12

are you familiar with the distributive property?

Answer 13

Yes, but i'm just sort of confused. Do you just multiply by x?

Answer 14

well when you use the distributive property, you go from something like

2*(3+5)

to

2*3 + 2*5


To factor, we just do it in reverse, so

we go from

2*3 + 2*5

to

2*(3+5)

Make sense?

Answer 15

So, if the problem was: x²-16x+64
I would go: (x²-4x)(-4x+64)?
Then what?

Answer 16

(x²-4x)+(-4x+64), but you're on the right track

Answer 17

oh wait, sry i misread

Answer 18

It should be (x²-8x)+(-8x+64)

Answer 19

I took -16 and cut it in half to get -8. So -16x is -8x-8x

Answer 20

So, then it would be: (x²-8x)+(-8x+64)?

Answer 21

yes

Answer 22

So, then I just multiply the groups by x²?

Answer 23

Now you factor x from x²-8x


x²-8x

x*x-x*8

x(x-8)

Answer 24

Ugh. I"m so lost.

Answer 25

Then you factor -8 from -8x+64

-8x+64

-8x+(-8)*(-8)

-8(x-8)


So

(x²-8x)+(-8x+64)

becomes

x(x-8) -8(x-8)

Answer 26

Alright, let's try it like this. How good are you at FOILing?

Answer 27

I'm familiar with the concept.
It's like First.Outside.Inside.Last. or something, right?

Answer 28

yes

Answer 29

Let's say we had (x+3)(x+5)

multiply First terms: x*x = x^2

multiply Outer terms: x*5 = 5x

multiply Inner terms: 3*x = 3x

multiply Last terms: 3*5 = 15


Now add up the terms we just got


x^2+5x+3x+15

x^2 + 8x + 15


Look familiar?

Answer 30

YES. Oh my goodness. So much easier. :)

Answer 31

lol great, so in that last post, I went from (x+3)(x+5) to x^2+8x+15


In the other posts above this, we're going backwards...so that may explain why it's hard to grasp.

Answer 32

Okay wait, let me see if I got this:

Answer 33

alright

Answer 34

x²-16x+64
It would go: x² x x²=
x² x -16x=
and x² x 64=
right?

Answer 35

ok think of it like this


when we factor x²-16x+64, we want to place it in the form (x+a)(x+b)...we basically want the two to be equal.


What do you get when you FOIL out (x+a)(x+b)?

Let's find out...

multiply First terms: x*x = x²

multiply Outer terms: x*b = bx

multiply Inner terms: a*x = ax

multiply Last terms: a*b = ab


Now add up the terms

x² + bx + ax + ab

x² + (a+b)x + ab


So

x²-16x+64 = (x+a)(x+b)

becomes

x²-16x+64 = x² + (a+b)x + ab


With me so far?

Answer 36

great patience Jim, keep up the good work!

Answer 37

So, it would look like:
x²-16x+64= x²?

Answer 38

I just suck at this so bad. You're trying so hard and i'm just not following it at all. I'm so sorry.. /:

Answer 39

You're fine, don't stress over this. Just take things one at a time. You got the FOIL idea down, so go from there.

Answer 40

Okay. So, one step at a time looks like this:

Answer 41

Okay. So then, how do I group them off so I can FOIL them?

Answer 42

wait a sec, you said "one step at a time looks like this:" but it's blank after that...I must be missing something

Answer 43

No, you're not. I was going to FOIL them.. But, they aren't group off into the inner and outer terms so I can't..right?

Answer 44

oh gotcha

Answer 45

When you FOIL, you're going from (x+3)(x+5) = x^2+8x+15

But....

when you factor, you're going in reverse. So you're not really FOILing, but you're using the same basic ideas

Answer 46

Forward (FOIL): (x+3)(x+5) to x^2+8x+15

Backward (Factor): x^2+8x+15 to (x+3)(x+5)


See the connection?

Answer 47

Yes, I do.
I think i'm having troubles with there being 3 numbers in my equations and only two in the examples you're showing me.

Answer 48

What do you mean as in 3 numbers?

Answer 49

My equations is: x²-16x+64
We've been trying to get me too understand how to factor and solve it but I can't figure it out..

Answer 50

Notice how in (x+3)(x+5) the 3 and 5 add to 8 AND 3 and 5 multiply to 15


So 3+5 = 8 AND 3*5 = 15


This is a shortcut to FOILing something like this

Answer 51

So if I gave you x^2+8x+15 and I asked you to factor, you can use this idea in reverse. You would ask yourself "what two numbers multiply to 15 AND add to 8?"

The answer would be "3 and 5" to give you the factorization (x+3)(x+5)

Answer 52

OH.

Answer 53

By golly, I think I get it.

Answer 54

Let me try!

Answer 55

that's great :)

Answer 56

alright

Answer 57

Wait, really quick can you set up the equation using my math problem?
x²-16x+64

Answer 58

sure thing

Answer 59

x²-16x+64 factors to something like (x+a)(x+b) where a and b are some numbers (we don't know yet), agreed?

Answer 60

Wait, why don't we know the numbers?

Answer 61

because we're going to find them

Answer 62

say we had x^2+8x+15 and we didn't know its factorization (yet), we would have to find those two numbers right?

Answer 63

Those two numbers would be 3 and 5 though, right?

Answer 64

Em you need to find two numbers that add up to 16 and when the same numbers are multiplied they equal 64

Answer 65

Well you know that because you saw (x+3)(x+5) already...so that's why you know it's 3 and 5. Does that make sense?

Answer 66

There aren't any that add and equal 16 that multiply to get 64.

Answer 67

So basically what I'm getting at is that with x²-16x+64, you don't know those two numbers yet...but you can find them if it can be factored


So what two numbers BOTH multiply to 64 AND add to -16?

Answer 68

There is! You can do it. Start simple by thinking of what adds up to 16

Answer 69

UGH. Omfg. I know that -4 x -16 = 64.

Answer 70

Or by what multiples to 64

Answer 71

Besides 4 and 16 what else?

Answer 72

8

Answer 73

so 8 is the first number, the second is ...?

Answer 74

8?

Answer 75

Yes and does 8+8 =16? YES!!!

Answer 76

you got it, so 8 and 8 add to 16 and multiply to 64

We want the 16 to be negative, so flip the sign of both '8's to get -8 and -8


So

-8 + (-8) = -16

-8*(-8) = 64

Answer 77

This means that x²-16x+64 factors to (x-8)(x-8)

Answer 78

Making it a perfect square I believe

Answer 79

yes, so you condense the terms to get (x-8)²

Answer 80

(x-8)²= x²-16x+64?

Answer 81

exactly

Answer 82

Sweet jesus. How much would you hate me if I asked if we could do another one to make sure I can do it on my own?

Answer 83

lol I wouldn't hate you at all

Answer 84

But you got all that we went through?

Answer 85

Go ahead bby gurl

Answer 86

Kind of.. I'm still a little foggy.

Answer 87

alright, well you can always scroll up and look over it again...or you can ask to go over a certain part again

Answer 88

Okay. Hold on. I need to find my pencil and get some water. :b

Answer 89

lol this is definitely a marathon (lol jk)

Answer 90

It really is though because I scuk at this stuff. :b

Answer 91

naw, you're doing fine

Answer 92

the next problem is: x²+11x+30.25

Answer 93

and you want this as a perfect square?

Answer 94

Yes.

Answer 95

Ok, another shortcut I should have mentioned earlier is this...


Step 1) Take the number in front of the x (without the exponent). This number is 11

Step 2) Cut the number in half. So 11 divided by 2 is 5.5

Step 3) Square 5.5 to get 5.5*5.5 = 30.25


Notice how starting with 11, and following these steps, leads us to 30.25


So this means that x²+11x+30.25 is a perfect square trinomial and it factors to (x+5.5)²


I'm not sure why I didn't think of this shortcut before, but the stuff we went over is still really important for factoring in general. So these two ideas are useful (one more than the other depending on what you're factoring)

Answer 96

Okay.. On #7 it says: Convert each quadratic function to vertex form. Give the coordinates of the vertex.
y=x²=8x
Help omg.

Answer 97

I'm going to use the shortcut I just explained above to factor x²-16x+64


Step 1) Take the number in front of the x (without the exponent). This number is -16.


Step 2) Cut this number in half to get -8.


Step 3) Square -8 to get 64.


So this shows us that x²-16x+64 is a perfect square. So it factors to (x-8)²

Answer 98

y=x²=8x is there a typo here?