Question

Are you ready for adventure? You bet you are!

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.

Answer 1

To open these doors, you must speak three functions in standard form.
One function, f(x), with two real rational solutions.
One function, g(x), with two real irrational solutions.
One function, h(x), with two complex solutions.

Create these three functions and explain to Professor McMerlock how you know these functions meet each condition. Remember, he is a Professor so use complete sentences. (Hint: Make sure that the b is even on g(x).)

Answer 2

@phi

Answer 3

the number of solutions a polynomial has is equal to the DEGREE (highest exponent on the variable). You want an equation with the highest exponent on x to be 2

Answer 4

see http://www.teacherschoice.com.au/Maths_Library/Algebra/Alg_6.htm

and
http://www.regentsprep.org/Regents/math/algtrig/ATE3/discriminant.htm

for f(x), you want the discriminant to be a perfect square (4, 9, 16, 25, etc)
for g(x), you want the discriminant to not be a perfect square... lots of choices
for h(x), you want the discriminant to be negative.

Each function will have the form
ax^2 + bx + c
where you pick the 3 numbers a, b and c so that b*b - 4*a*c fits the criteria

Answer 5

4x^2 + 9x + 16 would work for f(x)?

Answer 6

Alright I finished my other paper so now I can focus on this one and get it done :)

Answer 7

let's see
a=4, b=9, c=16

b*b - 4*a*c
9*9 - 4 *4*16
81 - 16*16
81-256
-175

First, it is negative. That means complex
(when you take the square root of a negative number, you get complex numbers)
also 175 is not a perfect square
13*13 is 169 and 14*14 is 196
the square root of 175 will be between 13 and 14.

Answer 8

hmm so we have to come up with a better set to get a perfect square...

Answer 9

Any suggestions?

Answer 10

one way is work backwards
say you want 36 as the perfect square

pick a to be a nice number, like 1!
b*b - 4*a*c = 36
a is 1, so
b*b -4*c = 36

b*b = 36+4*c

now play around with numbers for c

Answer 11

hmm.. How about 1 for c lol?

Answer 12

not that simple, because you have
b^2 = some number
you need to take the square root of both sides
b= sqr(number)
so you want 36+4c to be a perfect square

list the squares 49, 64, 81, 100
subtract 36 from each
if the remainder can be divided by 4, that is a winner

Answer 13

how about 64?

Answer 14

if 4c was 64, what is
36+4*c ?

Answer 15

36 + 64 = 100

Answer 16

or 64 - 36 = 28
/4 = 7?

Answer 17

100 - 49 = 51?

Answer 18

/4 is 12.75

Answer 19

let's start over

a=1,
we want b^2 - 4ac = 36 (36 is a perfect square)
b^2 -4c = 36
b^2 = 36+4c
we want 36+4c to be a perfect square. 36+64 = 100, which is good
that means we want 4c=64
what is c ?

Answer 20

c = 16

Answer 21

b^2 = 36+4*16= 32+64= 100
b= 10

now double check:
b^2 - 4 a c
with a=1, b=10, c=16
10*10 - 4*1*16
100 - 64
36
and when we take the square root of 36, we get 6.

so
\[ f(x)= x^2 +10x +16 \]
is a winner.

Answer 22

lol sweet :D

Answer 23

next lol

Answer 24

One function, g(x), with two real irrational solutions.

Answer 25

it just occurred to me there is an easier way:
we want a quadratic with nice roots, so pick them: x= -2 and x=-8 (these are the roots we would get for h(x))
x= -2
add +2 to both sides
x+2=0
similarly
x+8= 0
(x+2)(x+8) = 0
multiply this out
x^2 +10x +16

Answer 26

g(x) is easy
you have b^2 = 36+4c (with a=1)
pick any c that does not give a perfect square for 36+4c

Answer 27

uhh 3 lol?

Answer 28

try it (I am sure it works)

Answer 29

what is b^2 - 4 a c
with
a=1 , c= 3, b =10 ?

Answer 30

10^2 - 4(1)(3)
so... 100 - 12? which is 88

Answer 31

88/4 = 22

Answer 32

no divide by 4

the roots are
\[ x= \frac{-b±\sqrt{b^2 - 4 a c}}{2a} \]
you just found b^2 - 4 a c = 88
so the roots are
\[ x= \frac{-10±\sqrt{88}}{2} \]
you can factor 88 into 4*22 and simplify sqr(4)*sqr(22) to 2*sqr(22)
\[ x= \frac{-10±2\sqrt{22}}{2} \]
divide by 2 into both terms up top
\[ x= -5 ± \sqrt{22} \]

Answer 33

for the last part
you want
b^2 - 4 a c to be negative

just make a*c bigger than b^2
i.e. pick a smaller number for b, and big numbers for a and c

Answer 34

brb phone

Answer 35

I'm back