Question

Wow... I really need help on this problem. Its very overwhelming.

Answer 1

For this adventure, you and world renowned Professor Sherlock McMerlock are traveling to the Lost Island of Laplaya. When the boat arrives on the island shore, you and Professor McMerlock disembark on your adventure. You trudge through the jungles and arrive at three impressively large doors. About eye level on each door is an intricately carved keyhole. Directions are scratched into the wood above each keyhole.


1. To open these doors, you must match the number and type of solutions for the following two functions in standard form. •f(x) = x2 + 6x – 16
•g(x) = x2 +6x + 1



Match the following descriptions of the solutions to each of the functions above.
Hint: they each have their own match.
•Two real irrationals solutions
•Two real rationals solutions

After matching these functions, explain to Professor McMerlock how you know these functions meet each condition. Remember, he is a professor, so use complete sentences.



Like magic, the doors creak open with a burst of dust, musty air, and shrieking bats. You lead McMerlock down the first corridor. It is so dark that you can barely see your own outstretched hand. The darkness consumes you as you continue to trek deeper and deeper in the hallway. Suddenly, a mystical suit of armor blocks your path. A voice emanates from inside it.




2.“To pass by me, you must tell me how to convert standard form into the general, vertex form... I have a test on it next week.”

Explain how to convert f(x) into the general, vertex form of the equation. Use complete sentences.



Satisfied with his notes, the guardian lets you pass into the next chamber. As you enter the dimly lit room, McMerlock points out that the floor tiles have numbers on them, and that some floor tiles are missing. On the ceiling is painted a cryptic message.




3.“Only the solutions of g(x) will lead you safely across.”

Find the solutions of g(x). Show each step.

You can feel the treasure of the Lost Island of Laplaya in your grasp, as you deftly step on the correctly numbered stones. A wise old woman sits in front of the treasure. You can hear the crackle of magic coursing through her fingertips. The wise old woman grants you the treasure of the Lost Island of Laplaya! The treasure is an awesome ability to complete the square! Way to go! Professor McMerlock is thankful he picked you and promises to call on you again for another adventure.

Answer 2

In general, we have

ax^2 + bx + c = 0

the discriminant D is given by

D = b^2 - 4ac

Answer 3

If D > 0, then you will have 2 different real solutions

If D = 0, then you'll have exactly one real solution

If D < 0, then you won't have any real solutions at all (they will both be complex)

Answer 4

1. To open these doors, you must match the number and type of solutions for the following two functions in standard form. •f(x) = x2 + 6x – 16
•g(x) = x2 +6x + 1


what do I even do.. I don't really even know what the first part is asking me to do

Answer 5

If D > 0 and if D is a perfect square, then you'll have 2 rational (ie fractional) solutions

If D > 0 and if D is not a perfect square, then you'll have 2 irrational solutions

Answer 6

@Jhannybean

Answer 7

For example

say we had 2x^2 + 6x - 12

we can see that

a = 2, b = 6 and d = -12

plug these values into the discriminant formula to get


D = b^2 - 4ac

D = (6)^2 - 4(2)(-12)

D = 36 + 96

D = 132


This value of D is positive. So we have 2 real solutions

The value D = 132 is NOT a perfect square, so we have 2 irrational solutions

Answer 8

I have no idea what youre talking about honestly.

Answer 9

have you seen ax^2 + bx + c = 0 before?

Answer 10

yes

Answer 11

how about D = b^2 - 4ac

Answer 12

yeah

Answer 13

so I made up an example, I just randomly came up with 2x^2 + 6x - 12

Answer 14

I then could see that a = 2, b = 6, & c = -12

Answer 15

are you suppose to make something up?

Answer 16

no I made something up as an example to show how it all works

Answer 17

I guess it'd be easier to use the actual functions you were given

Answer 18

ok so let's say we had

x^2 + 6x - 16

which is the same as

1x^2 + 6x - 16

Answer 19

match up 1x^2 + 6x - 16 to ax^2 + bx + c

what are the values of a,b,c ?

Answer 20

a is 1 b is 6 and c is -16

Answer 21

correct

Answer 22

you then plug those values into D = b^2 - 4ac

Answer 23

D = b^2 - 4ac

D = (6)^2 - 4*(1)*(-16)

D = 36 + 64

D = 100

Since D is positive we have 2 real solutions
Since D is a perfect square, we have 2 rational solutions (the two rational solutions are the 2 real solutions)

Hopefully that makes sense?

Answer 24

yes that does...

Answer 25

you'll do the same for x^2 + 6x + 1

Answer 26

d is 32?

Answer 27

correct, so what type of solutions does x^2 + 6x + 1 have ?

Answer 28

positive? im not sure about this part.

Answer 29

You'll use these rules


If D > 0 and if D is a perfect square, then you'll have 2 rational (ie fractional) solutions.

If D > 0 and if D is not a perfect square, then you'll have 2 irrational solutions.

Answer 30

so, is D a perfect square?

Answer 31

ik that the correct answer is no but im not really sure why.

Answer 32

32 is not a perfect square because there is no whole number you can square to get 32

in other words,

x^2 = 32

has no solution x where x is a whole number

Answer 33

a number like 25 is a perfect square because

5^2 = 5*5 = 25

but you can't do something similar with 32

Answer 34

oh okay I get it.. so now I know how to explain

Answer 35

2.“To pass by me, you must tell me how to convert standard form into the general, vertex form... I have a test on it next week.”

Answer 36

So because D = 32 is not a perfect square, this means that x^2 + 6x + 1 has 2 irrational solutions

Answer 37

did you want to convert x^2 + 6x + 1 into vertex form?

Answer 38

yes what is the formula

Answer 39

ok first we need to get the x coordinate of the vertex

Answer 40

we will use the formula

x = -b/(2a)

the 'a' and 'b' come from ax^2 + bx + c = 0

Answer 41

so f(x) = x^2 + 6x – 16

and we know

a = 1
b = 6
c = -16

Answer 42

plug a = 1 and b = 6 into x = -b/(2a) to get


x = -b/(2a)

x = -6/(2*1)

x = -3


with me so far?

Answer 43

yes.. that's the answer I got

Answer 44

that is the x coordinate of the vertex

Answer 45

you will use this to find the y coordinate of the vertex

Answer 46

y = x^2 + 6x - 16

y = (-3)^2 + 6(-3) - 16 ... plug in x = -3

y = 9 + 6(-3) - 16

y = 9 - 18 - 16

y = -25

therefore, the y coordinate of the vertex is y = -25

Answer 47

all together, the vertex is the point (-3, -25)

Answer 48

so (h,k) = (-3, -25)

h = -3
k = -25

Answer 49

from y = x^2 + 6x - 16, we know that a = 1

Answer 50

you will plug these 3 items

a = 1, h = -3, k = -25

into this formula

y = a(x-h)^2 + k

Answer 51

So...

y = a(x-h)^2 + k

y = 1(x-(-3))^2 + (-25)

y = (x+3)^2 - 25

-------------------------------------------------------

This means y = x^2 + 6x - 16 turns into y = (x+3)^2 - 25

y = (x+3)^2 - 25 is in vertex form. It's called vertex form because you can easily pick out the vertex from this form of the equation.

Answer 52

I need help with the last step
3.“Only the solutions of g(x) will lead you safely across.”

Find the solutions of g(x). Show each step.

You can feel the treasure of the Lost Island of Laplaya in your grasp, as you deftly step on the correctly numbered stones. A wise old woman sits in front of the treasure. You can hear the crackle of magic coursing through her fingertips. The wise old woman grants you the treasure of the Lost Island of Laplaya! The treasure is an awesome ability to complete the square! Way to go! Professor McMerlock is thankful he picked you and promises to call on you again for another adventure.

Answer 53

@strawberryswing

Answer 54

Wow