Question

find three consecutive integers such that the product of the first and the second is equal to the product -6 and the third

Answer 1

does
anybody have an idea

Answer 2

yes

Answer 3

so do you know what to do? is there a technique

Answer 4

One approach is to try choosing a number, and try it with that and the next 2 consecutive integers and see if it works. That may sound like guessing...

Answer 5

is there another way?

Answer 6

Another approach/technique is to use variables and set up an equation.

Notice that consecutive integers look like this:
if first integer is "n",
then 2nd integer is "n + 1" and third integer is "n + 2"

Answer 7

@zordoloom do you know?

Answer 8

So: find three consecutive integers such that the product of the first and the second is equal to the product -6 and the third

product of first and second integers: (n) * (n + 1)
product of -6 and third integer: (-6) * (n + 2)

Those two products are supposedly equal:

n(n+1) = (-6)(n + 2)

Multiply everything out, simplify, and solve for "n". Then the three consecutive integers are that "n", and the next two integers after.

Answer 9

so i would just solve it? how do you know that the 2 "n's" are equal. the two integers could be different

Answer 10

Say, for example, that I know "n" is equal 2.

Then what is the next consecutive integer? It's 3, right? But it's also just "n + 1"

And the third consecutive integer is 4, but you can also write it as "n + 2"

I only used one variable... the other integers are just expressed using that same "n", so the n's are all the same... there's only one "n".

Answer 11

o ok thanks for clarification. so now i would solve for n(n+1) = (-6)(n + 2)

Answer 12

how do you solve it? :/ n^2 + n = -6n - 12?

Answer 13

yes :)

Don't worry... I understand your concern about the using the "n" terms. But this is a decent technique that looks better (from a teacher's perspective) than guessing! :)

Answer 14

what @AriPotta is saying is making sense what should i do next after i distribute

Answer 15

n(n+1) = (-6)(n + 2)

Multiply both sides out first, then collect terms...
n(n+1) = (-6)(n + 2)
n^2 + n = -6n - 12
n^2 + n + 6n + 12 = -6n + 6n - 12 + 12 (get everything on left)
n^2 + 7n + 12 = 0

Answer 16

then would i
n^2+7n=-12
n^2+n =-12/7
??????????/

Answer 17

is this...factoring? i haven't learned much about factoring, but this looks like where you would factor :l

Answer 18

o yeah true i forgot about factoring

Answer 19

Hmm... no, you need to factor instead.

Answer 20

i haven't learnt it that well

Answer 21

ill try

Answer 22

n^2 + 7n + 12 = 0
What are two numbers you can multiply to get 12 that when you add them, you get 7?

Answer 23

(n+4) (n+3)

Answer 24

is that the answer

Answer 25

no, you still have to find the numbers?

Answer 26

Yes, (n + 4)(n + 3) = 0

so, that expression is true if n = -4 or if n = -3... so you actually have two values for n.

Answer 27

:O wat

Answer 28

The final step should be to check both those solutions for n and see if they work in the original problem.

n = -3
next consecutive integer: -2
third consecutive integer: -1
and (-3)(-2) = (-6)(-1) <<--- Yay, true!

n = -4
next consecutive integer: -3
third consecutive integer: -2
and (-4)(-3) = (-6)(-2) <<<--- also true, yay!

So the integers could be -3, -2, -1 OR -4, -3, -2

Answer 29

thanks

Answer 30

ooh

Answer 31

wait, how'd you know it was negative?

Answer 32

The factored form was (n + 4)(n + 3) = 0

When you look at those 2 sets of parentheses, ask yourself what n value would make each one = 0?

The reasoning is:

A * B = 0
if either A or B is = 0, then A * B is equal 0.
So, if A = (n + 4)
and B = (n + 3)

What value of n makes A = 0? What value of n makes B = 0?

Answer 33

ooooooooooooooh. i get it. thanks :D

Answer 34

So, @AriPotta, that's why you try to factor in a problem like this... it makes it quick to get the solutions.

If you have (x - 5)(x + 6) = 0, then the solutions are x = 5, x = -6... and so on.

Answer 35

The original equation that would have led to that factoring is:

x^2 + x - 30 = 0

(x - 5)(x + 6) = 0

Solutions: x = 5, x = -6

Answer 36

if i wasn't your fan already, i'd fan ye

Answer 37

thanks ;)

Answer 38

A Mathematica solution is attached. The general idea should be apparent.