Question

√2x+8 - √x+11 = -1

find the solution set

Answer 1

Is the 2 and x the only ones in sq root?

Answer 2

(2x+*) is one sqrt and (x+11) is another one :)

Answer 3

ok, I just about got it

Answer 4

okay i go the answer as 0 but apparently is incorrect

Answer 5

my guess is 14 using no algebra

Answer 6

so you rearranged sides and went from there?

Answer 7

yeah, i squared both sides and used FOIL but idk

Answer 8

to hell with foil

Answer 9

lets be smart

Answer 10

agreed but thats how i was taught D:

Answer 11

inside the radicals are \(2x+8\) and \(x+11\) right?

Answer 12

yesssss

Answer 13

ok lets focus on \(x+11\)
in order for it to be a perfect square like say \(16\) or \(25\) \(x\) what could \(x\) be?

Answer 14

and the answer is, it could be \(5\) since \(5+11=16\)
BUT \(2\times 5+8=18\) is NOT a perfect square, so we toss that out

Answer 15

x=14? idk if thats what you mean

Answer 16

so what if \(x+11=25\)? then \(x=14\) and
\[2\times 14+8=36\] another perfect square !! got it on the second try

Answer 17

the check is easy
\[\sqrt{2\times 14+8}-\sqrt{14+11}=\sqrt{36}-\sqrt{25}=6-5=1\]

Answer 18

ooh snap !! you want \(-1\) right !!

Answer 19

superb. then the solution is 14! thank you, i always mess up with the foil method D;

Answer 20

no no i was wrong!!

Answer 21

wait... why D: it looks right?

Answer 22

you want \(-1\) not \(1\) i read it wrong

Answer 23

oh okay. so what would i do then?

Answer 24

so lets make \(x+11=9\) another perfect square

Answer 25

that makes \(x=-2\) and it makes
\[2x+8=2\times (-2)+8=4\] that's better

Answer 26

now we get it
\[\sqrt{2\times -1+8}-\sqrt{-2+11}=\sqrt4-\sqrt9=2-3=-1\]

Answer 27

it is always better to think than it is to use algebra
algebra is for math teachers, thinking is for humans
but if you want to do it the math teacher way we can do that too

Answer 28

would there be a solution set?

Answer 29

more then just 8?

Answer 30

yes, it is \(\{-2\}\)

Answer 31

okay, now can you explain to me how you got to find that answer so quickly?

Answer 32

yes, that is what i was trying to do
let me say it in plain english

Answer 33

you have two radicals
inside one radical is \(2x+8\) and inside the other is \(x+11\)

Answer 34

in order for the answer to come out to a nice whole number, the number inside BOTH radicals has to be a perfect square, like
\[1,4,9,16,25,36,49,64,...\]

Answer 35

how likely is that? not very
the problem has to be cooked up carefully to make it work
so i made a guess
i guessed that \(x+11=16\) making \(x=5\) but that did not make \(2x+8\) a perfect square, it made it 18
toss that out

Answer 36

then i tried \(x+11=25\) making \(x=14\) and was happy because \(2x+8=36\) but i was again mistaken because that gave \(6-5=1\) and you wanted \(-1\)
too big
try \(x+11=9\) which makes \(x=-2\) and lo and behold \(2x+8=4\) done

Answer 37

now we can do it the donkey way if you like
start with
\[\sqrt{2x+8}-\sqrt{x+11}=-1\] then add \(\sqrt{x+11}\)to both sides giving
\[\sqrt{2x+8}=\sqrt{x+11}-1\] then square both sides
\[2x+8=x+11-2\sqrt{x+11}+1\] ick

Answer 38

now we have to put all the stuff not in the racial on the left and square again!!

Answer 39

plus it is easy to make a mistake when squaring \((\sqrt{x+11}-1)^2\)

Answer 40

okay! im going to take note of this! thank you so much!!!! :D

Answer 41

we can do it if you like, we will still get \(x=-2\)

Answer 42

and if you cannot guess correctly i spose you have to do it the math teacher way, but try to make a good guess first
especially if you have two radicals

yw