Question

MEDAL FOR HELP AYYYE LMAO



Find 4 rational numbers such that the product of the second,third, and fourth number = -63. The second number is 2 more than the first number, the third is 5 less than twice the second, and the fourth is six less than the first.

Answer 1

Ok so let's say we have the rational numbers a,b,c, and d

We are given:
The product of b,c and d is -63.
b is 2 more than a.
c is 5 less than twice the b.
d is 6 less than a .

Do you know how to translate the givens into mathematical equations.

Answer 2

The answer is
\[{ -3, -1, -7, -9}\]
\[{2.5, 4.5, 4, -3.5}\]
\[{5, 7, 9, -1}\]

Answer 3

yeah

Answer 4

ok so i got the first is x, the second is x + 2, the third is 2x-1, and the fourth x-6

Answer 5

idk what to do after that

Answer 6

how do you know they are in that form?

Answer 7

What do you mean?

Answer 8

ok so i got the first is x, the second is x + 2, the third is 2x-1, and the fourth x-6
how did you find this out?

Answer 9

ok if the second is 2 more than the first (x) then its x +2

Answer 10

then the third is 2(x+2) -5

Answer 11

=2x-1

Answer 12

So are you saying that it is given the the first, second, third, and fourth numbers they are talking is x,x+2,2x-1,x-6?

Answer 13

the 4th is six less than the first (x) -6

Answer 14

I think

Answer 15

yeah its given :P

Answer 16

do u then set it up into a polynomial? (x+2)(2x-1)(x-6) = -63 ?

Answer 17

ok
so...
Ok so let's say we have the rational numbers a,b,c, and d

We are given:
The product of b,c and d is -63.
b is 2 more than a.
c is 5 less than twice the b.
d is 6 less than a .

Do you know how to translate the givens into mathematical equations.

instead of me writing a,b,c, and d
then we can write x,x+2,2x-1,x-6

So you are write that first sentence translates to: (x+2)(2x-1)(x-6)=-63

So let's look at solving: (x+2)(2x-1)(x-6)=-63

Answer 18

ok

Answer 19

I just copied paste some of that so I didn't have to look back up

Answer 20

thats cool

Answer 21

This is a cubic equation

Answer 22

so it simplifies to \[2x^3-11x^2-8x+75 = 0\]

Answer 23

let me check

Answer 24

i may have miscalculated but thats what I got

Answer 25

two terms are off

Answer 26

but you did good on the first and last term on the left hand side

Answer 27

\[(x+2)(2x-1)(x-6)=-63 \\ (x+2)(2x-1)(x-6)+63=0 \\ [(x+2)(x-6)](2x-1)+63=0\]
let's look at multiplying first (x+2) and (x-6)

Answer 28

\[(x^2-4x-12)(2x-1)+63=0\]
see if you can (x^2-4x-12) and (2x-1)

Answer 29

\[2x^3-x^2-8x^2+4x-24x+12\]

Answer 30

?

Answer 31

that actually looks lovely and then we have +63 to tact on

Answer 32

\[2x^3-9x^2-20x+75\]

Answer 33

\[2x^3-9x^2-20x+75=0\]

Answer 34

now let's see if we can apply the rational root theorem

Answer 35

ooooooohhhh

Answer 36

ok i got it

Answer 37

that is we look at the factors of 75 over the factors of 2

\[\pm \frac{75}{2}, \pm\frac{25}{2}, \pm \frac{15}{2}, \pm \frac{5}{2}, \pm \frac{3}{2}, \pm \frac{1}{2}, \pm \frac{75}{1}, \pm \frac{25}{2}, \pm \frac{15}{1}, \pm \frac{5}{1}, \pm \frac{3}{1}, \pm \frac{1}{1}\]
that is a lot of numbers to check :p

Answer 38

but you got it you say?

Answer 39

Yeah our teacher allows graphing calculators

Answer 40

lol oh that makes it a tad easier

Answer 41

yeaaahh XD thanks for the help!

Answer 42

np