Videos > Extraneous Solutions for Radical Equations - Algebra, Mathematics
Extraneous Solutions for Radical Equations - Algebra, Mathematics Video Lecture
FAQs on Extraneous Solutions for Radical Equations - Algebra, Mathematics Video Lecture
| 1. What are extraneous solutions in radical equations? |  |
Ans. Extraneous solutions are solutions that arise when solving radical equations but do not actually satisfy the original equation. They can occur when squaring both sides of an equation or when solving equations with even roots.
| 2. Why do extraneous solutions occur in radical equations? | |
Ans. Extraneous solutions occur in radical equations because squaring both sides of an equation can introduce additional solutions that do not satisfy the original equation. This happens because squaring a negative number yields a positive result, which may not be a valid solution for the original equation.
| 3. How can we identify extraneous solutions in radical equations? | |
Ans. To identify extraneous solutions in radical equations, one must check the solutions obtained by substituting them back into the original equation. If any of the solutions do not satisfy the original equation, they are considered extraneous.
| 4. Can extraneous solutions occur in all types of radical equations? | |
Ans. Extraneous solutions can occur in radical equations involving even roots (such as square roots) or when squaring both sides of an equation. They do not typically occur in equations with odd roots (such as cube roots).
| 5. How can we avoid or minimize extraneous solutions in radical equations? | |
Ans. To avoid or minimize extraneous solutions in radical equations, it is important to carefully analyze each step of the solution process. Avoid squaring both sides of an equation unless necessary and remember to check each obtained solution by substituting it back into the original equation to ensure its validity. Additionally, it is important to understand the domain restrictions of the original equation to avoid introducing extraneous solutions.
Text Transcript from Video
we're asked to solve to X is equal to
the principal square root of 14 plus 10x
plus 2 so whenever you see one of these
equations that have a variable we need
to solve for under the radical sign what
you want to do is you want to isolate
the radical sign and then square both
sides essentially to get rid of it but
you have to be very careful when you do
that because when you square radical
sign you actually lose the information
then you were taking the principal or
the positive square root and so when you
get your final answers you've got to
make sure that they work because some of
the answers might have been based on
taking the negative square root and just
to make sure that I guess the best way
to see that tangible is is to try to do
this problem and let's see if that shows
up here so the first thing we want to do
is isolate this radical sign best way to
do that is to get rid of this two here
the best way to do that is to subtract
two from the right but if we do it from
the right we also have to do it from the
left and so our equation becomes on the
left hand side we have 2x minus 2 and
then on the right hand side that is
equal to the square root of 14 plus 10x
and these guys right over there
cancel out and so now we can square both
sides so now we can square both sides of
this equation and this is the step where
we're losing information because when we
square the right hand side right over
here when we square the right hand side
and this is the whole point of squaring
it we're just going to get 14 we're
going to get 14 plus 10x and the reason
why we lost information is this we would
have also gotten this if this was the
negative square root but if this was a
negative square root this would've been
a fundamentally different problem you
would have had a negative square root
out here but we're getting the same
thing over here so one of our answers
might actually satisfy this equation as
opposed to the one that we need to solve
which is the one that doesn't have the
negative over there so the best way to
do it is just to salt square solve and
then see which of these work and which
of them don't by substituting back in
the original equation so we get 14 plus
10x on the right hand side on the left
hand side and we've done this many times
in previous videos if you square
something if you have a plus B squared
this is equal to a squared plus 2 a B
plus B squared you can do the
distributive property twice
expanding this out and multiplying or
you can do foil if you like that better
but it is good to recognize the special
case of squaring a binomial it fits in
this pattern so 2x squared the whole
quantity is going to be 4x squared so if
you take 2x and I squared the whole
thing I get 4x squared and then I'm
going to take the product of both of
these terms so 2x times negative 2 is
negative 4x negative 4x times 2 is
negative negative 8x let me be clear how
I'm getting these so this is this term
right here is 2x squared this term over
here this term over here is 2 times 2x
times negative 2 so it's equivalent to
this term right over here and then
finally I'm gonna have to square this
right over here so negative 2 squared is
just positive 4 so this is analogous to
this term right over there and now we
have a quadratic equation and we can
simplify it so it looks like a
traditional quadratic equation so we can
do is get rid of this 14 plus 10x on the
right hand side once again best way to
do that subtract 14 from both sides
subtract 14 from both sides and subtract
10x from both sides and subtract 10x
from both sides and we are left with the
right-hand side is easy 14 minus 14 - 0
10x minus 10x is 0 we're just left with
a 0 on the left-hand side we have 4x
squared negative 8x minus 10x is
negative 18x and then 4 minus 14 4 minus
14 is negative 10 and that is going to
be equal to 0 and everything here is
divisible by 2 and so what we can do is
just to simplify this a little bit as
multiply both sides of this equation is
multiply both sides of this equation by
1/2 so let's multiply both which is the
same thing as dividing it by 2
well dividing both sides by 2 or
multiplying both sides by 1/2 and what
that does on the left-hand side 1/2
times 4 is 2 x squared or 1/2 times 4 x
squared is 2x squared 1/2 times negative
18x is negative 9x and then 1/2 times
negative 10
is negative five and on the right hand
side we have this is equal to zero and
so we can let's see if we can factor
this out we can try factoring by
grouping and when you factor by grouping
we're looking for two numbers so we're
looking for two numbers a let's call
them a and B that add up to negative
nine because we're going to break up
this term right here into ax and BX so
we they need to add up to negative nine
and their product needs to be the
product of two times negative five so a
times B needs to be equal to two times 2
times negative 5 or negative 10 and
since their product is a negative number
one of these has to be positive and one
of these has to be negative and we've
covered this and we've shown why this
works and I think many many many videos
ago but you might want to look that up
if you think I'm doing some type of
magic voodoo right over here so let's
think about numbers that that add up to
negative nine and whose product is
negative ten so if you look at 1 and 10
if you make one of these negative and
one of these positive you will either
get positive or negative 9 and so that
looks pretty good so if you have 1 and
negative 10 they add up to negative 9
and you take the product you get when
you take the product you get negative 10
1 times negative 10 is negative 10 so
let's break up this negative 9x into
let's break it up into a positive 1x and
a negative and a negative 10x so all I
did obviously 1 X minus 10x is negative
9x and that's why you see why they have
these coefficients have to add up to
negative 9 and then we have the 2 x
squared out here and then we have a
minus 5 out here and you should just see
this negative 9x is the same thing as I
wrote in green right over here and if
you don't like this factoring by
grouping the other option really is to
use the quadratic formula and you'll
just get the answers but this is a
little bit cleaner especially if you
have a gut feeling and the people have
written this problems seem to like to
give us integer solutions or close to
integer solutions so if you have a gut
feeling it's going to be one of those
factoring by grouping it's not a bad
idea and so we group we can group these
two guys and then we can group and then
we can group these two guys so we could
call this plus negative 10x so that we
can group it just like that
and so in this
first expression right here we can
factor out an X so this becomes x times
2x plus 1 that's this right over here
and then the second term over here we
can factor out it looks like we can
factor out a negative 5 so you factor
out a negative 5 you get negative 5
negative 10x divided by negative 5 is 2x
negative 5 divided by negative 5 is just
1 and now we can factor out a 2x plus 1
from both of these terms and so I'll do
that over here so if you factor out a 2x
plus 1 you get 2x plus 1 times you have
this X over here and then you have this
minus 5 over here and if this isn't
clear what I did here I just literally
factor these guys out or another way to
think is if you multiply if you
distributed this times that and if you
did if you multiply 2x times 2x plus 1
times X and 2x plus 1 times negative 5
this is exactly what you would get right
over here so another way of thinking of
this factoring this out is that we
undistributed it I guess is one way to
think about it but anyway this equation
is now 2x plus 1 times X minus 5 is
equal to 0 and obviously if you have the
product of two numbers and they equal to
0 that means one or both of them need to
be equal to 0 so we get either 2x plus 1
is equal to 0 or or is could be + or X
minus 5 is equal to 0 when you solve
this you get you could subtract 1 from
both sides you get 2x is equal to
negative 1 divide both sides by 2 you
get X is equal to negative 1/2 or X X is
actually it's not and or it's just or
either one of these will satisfy if
either one of these are equal to 0 then
this will satisfy the equation although
both of them can be and so and then or
satisfies that obviously so if and then
for here X minus 5 equals 0 add 5 to
both sides you get X I'll write it down
here you get X is equal to 5 or X is
equal to 5 now remember you can't just
say that these are the solutions because
one of these solutions might be for the
case where this was the negative square
root so we have to substitute back into
our region
equation and see if they work so first
let's try the situation with X is equal
to negative 1/2 if you do that you get
I'll do it up here I'm running out of
real estate I'll do it up here so let's
try the situation when X is equal to
negative 1/2 then this equation becomes
2 times negative 1/2 is equal to the
principal square root the principal
square root the principal square root of
14 plus 10 times negative 1/2 10 times
negative 1/2 plus 2 and so we get
negative 1 is equal to 10 times negative
1/2 is negative 5 14 times negative 5 14
times
sorry 14 plus negative 5 is 9 the
principal square root of 9 is 3 so we
get 3 plus 2 which is clearly not the
case this does not work
negative 1/2 does not work this is an
extraneous solution and you're like wait
why did that even show up when we did
the math because if you think about it
if this was the negative square root if
this equation happened to be have a
negative square root over here then this
would be a negative 3 and then it would
work and then it would work so this was
the solution for when we were taking the
negative square root but this is the
principal square root and so negative
1/2 does not work so let's try X is
equal to 5 I'll do it right over here so
if X is equal to 5 then we get 2 times 5
so that's 10 should be equal to the
principal square root of 14 plus 10
times 5 well 10 times 5 is 50 plus 14 is
64 plus 2 this is clearly true the
principal square root of 64 is 8 8 plus
2 is 10 so our answer for this radical
equation is X is equal to 5 it is not x
equals a negative 1/2 this is extraneous
the principal square root of 14 plus 10x
plus 2 so whenever you see one of these
equations that have a variable we need
to solve for under the radical sign what
you want to do is you want to isolate
the radical sign and then square both
sides essentially to get rid of it but
you have to be very careful when you do
that because when you square radical
sign you actually lose the information
then you were taking the principal or
the positive square root and so when you
get your final answers you've got to
make sure that they work because some of
the answers might have been based on
taking the negative square root and just
to make sure that I guess the best way
to see that tangible is is to try to do
this problem and let's see if that shows
up here so the first thing we want to do
is isolate this radical sign best way to
do that is to get rid of this two here
the best way to do that is to subtract
two from the right but if we do it from
the right we also have to do it from the
left and so our equation becomes on the
left hand side we have 2x minus 2 and
then on the right hand side that is
equal to the square root of 14 plus 10x
and these guys right over there
cancel out and so now we can square both
sides so now we can square both sides of
this equation and this is the step where
we're losing information because when we
square the right hand side right over
here when we square the right hand side
and this is the whole point of squaring
it we're just going to get 14 we're
going to get 14 plus 10x and the reason
why we lost information is this we would
have also gotten this if this was the
negative square root but if this was a
negative square root this would've been
a fundamentally different problem you
would have had a negative square root
out here but we're getting the same
thing over here so one of our answers
might actually satisfy this equation as
opposed to the one that we need to solve
which is the one that doesn't have the
negative over there so the best way to
do it is just to salt square solve and
then see which of these work and which
of them don't by substituting back in
the original equation so we get 14 plus
10x on the right hand side on the left
hand side and we've done this many times
in previous videos if you square
something if you have a plus B squared
this is equal to a squared plus 2 a B
plus B squared you can do the
distributive property twice
expanding this out and multiplying or
you can do foil if you like that better
but it is good to recognize the special
case of squaring a binomial it fits in
this pattern so 2x squared the whole
quantity is going to be 4x squared so if
you take 2x and I squared the whole
thing I get 4x squared and then I'm
going to take the product of both of
these terms so 2x times negative 2 is
negative 4x negative 4x times 2 is
negative negative 8x let me be clear how
I'm getting these so this is this term
right here is 2x squared this term over
here this term over here is 2 times 2x
times negative 2 so it's equivalent to
this term right over here and then
finally I'm gonna have to square this
right over here so negative 2 squared is
just positive 4 so this is analogous to
this term right over there and now we
have a quadratic equation and we can
simplify it so it looks like a
traditional quadratic equation so we can
do is get rid of this 14 plus 10x on the
right hand side once again best way to
do that subtract 14 from both sides
subtract 14 from both sides and subtract
10x from both sides and subtract 10x
from both sides and we are left with the
right-hand side is easy 14 minus 14 - 0
10x minus 10x is 0 we're just left with
a 0 on the left-hand side we have 4x
squared negative 8x minus 10x is
negative 18x and then 4 minus 14 4 minus
14 is negative 10 and that is going to
be equal to 0 and everything here is
divisible by 2 and so what we can do is
just to simplify this a little bit as
multiply both sides of this equation is
multiply both sides of this equation by
1/2 so let's multiply both which is the
same thing as dividing it by 2
well dividing both sides by 2 or
multiplying both sides by 1/2 and what
that does on the left-hand side 1/2
times 4 is 2 x squared or 1/2 times 4 x
squared is 2x squared 1/2 times negative
18x is negative 9x and then 1/2 times
negative 10
is negative five and on the right hand
side we have this is equal to zero and
so we can let's see if we can factor
this out we can try factoring by
grouping and when you factor by grouping
we're looking for two numbers so we're
looking for two numbers a let's call
them a and B that add up to negative
nine because we're going to break up
this term right here into ax and BX so
we they need to add up to negative nine
and their product needs to be the
product of two times negative five so a
times B needs to be equal to two times 2
times negative 5 or negative 10 and
since their product is a negative number
one of these has to be positive and one
of these has to be negative and we've
covered this and we've shown why this
works and I think many many many videos
ago but you might want to look that up
if you think I'm doing some type of
magic voodoo right over here so let's
think about numbers that that add up to
negative nine and whose product is
negative ten so if you look at 1 and 10
if you make one of these negative and
one of these positive you will either
get positive or negative 9 and so that
looks pretty good so if you have 1 and
negative 10 they add up to negative 9
and you take the product you get when
you take the product you get negative 10
1 times negative 10 is negative 10 so
let's break up this negative 9x into
let's break it up into a positive 1x and
a negative and a negative 10x so all I
did obviously 1 X minus 10x is negative
9x and that's why you see why they have
these coefficients have to add up to
negative 9 and then we have the 2 x
squared out here and then we have a
minus 5 out here and you should just see
this negative 9x is the same thing as I
wrote in green right over here and if
you don't like this factoring by
grouping the other option really is to
use the quadratic formula and you'll
just get the answers but this is a
little bit cleaner especially if you
have a gut feeling and the people have
written this problems seem to like to
give us integer solutions or close to
integer solutions so if you have a gut
feeling it's going to be one of those
factoring by grouping it's not a bad
idea and so we group we can group these
two guys and then we can group and then
we can group these two guys so we could
call this plus negative 10x so that we
can group it just like that
and so in this
first expression right here we can
factor out an X so this becomes x times
2x plus 1 that's this right over here
and then the second term over here we
can factor out it looks like we can
factor out a negative 5 so you factor
out a negative 5 you get negative 5
negative 10x divided by negative 5 is 2x
negative 5 divided by negative 5 is just
1 and now we can factor out a 2x plus 1
from both of these terms and so I'll do
that over here so if you factor out a 2x
plus 1 you get 2x plus 1 times you have
this X over here and then you have this
minus 5 over here and if this isn't
clear what I did here I just literally
factor these guys out or another way to
think is if you multiply if you
distributed this times that and if you
did if you multiply 2x times 2x plus 1
times X and 2x plus 1 times negative 5
this is exactly what you would get right
over here so another way of thinking of
this factoring this out is that we
undistributed it I guess is one way to
think about it but anyway this equation
is now 2x plus 1 times X minus 5 is
equal to 0 and obviously if you have the
product of two numbers and they equal to
0 that means one or both of them need to
be equal to 0 so we get either 2x plus 1
is equal to 0 or or is could be + or X
minus 5 is equal to 0 when you solve
this you get you could subtract 1 from
both sides you get 2x is equal to
negative 1 divide both sides by 2 you
get X is equal to negative 1/2 or X X is
actually it's not and or it's just or
either one of these will satisfy if
either one of these are equal to 0 then
this will satisfy the equation although
both of them can be and so and then or
satisfies that obviously so if and then
for here X minus 5 equals 0 add 5 to
both sides you get X I'll write it down
here you get X is equal to 5 or X is
equal to 5 now remember you can't just
say that these are the solutions because
one of these solutions might be for the
case where this was the negative square
root so we have to substitute back into
our region
equation and see if they work so first
let's try the situation with X is equal
to negative 1/2 if you do that you get
I'll do it up here I'm running out of
real estate I'll do it up here so let's
try the situation when X is equal to
negative 1/2 then this equation becomes
2 times negative 1/2 is equal to the
principal square root the principal
square root the principal square root of
14 plus 10 times negative 1/2 10 times
negative 1/2 plus 2 and so we get
negative 1 is equal to 10 times negative
1/2 is negative 5 14 times negative 5 14
times
sorry 14 plus negative 5 is 9 the
principal square root of 9 is 3 so we
get 3 plus 2 which is clearly not the
case this does not work
negative 1/2 does not work this is an
extraneous solution and you're like wait
why did that even show up when we did
the math because if you think about it
if this was the negative square root if
this equation happened to be have a
negative square root over here then this
would be a negative 3 and then it would
work and then it would work so this was
the solution for when we were taking the
negative square root but this is the
principal square root and so negative
1/2 does not work so let's try X is
equal to 5 I'll do it right over here so
if X is equal to 5 then we get 2 times 5
so that's 10 should be equal to the
principal square root of 14 plus 10
times 5 well 10 times 5 is 50 plus 14 is
64 plus 2 this is clearly true the
principal square root of 64 is 8 8 plus
2 is 10 so our answer for this radical
equation is X is equal to 5 it is not x
equals a negative 1/2 this is extraneous
About this Video
Oct 05, 2025
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