yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Solving square-root equations: two solutions | Mathematics III | High School Math | Khan Academy


3m read
·Nov 11, 2024

Let's say that we have the equation ( 6 + 3w = \sqrt{2w + 12} + 2w ).

See if you can pause the video and solve for ( w ), and it might have more than one solution, so keep that in mind.

All right, now let's work through this together. The first thing I'd like to do whenever I see one of these radical equations is just isolate the radical on one side of the equation. So let's subtract ( 2w ) from both sides. I want to get rid of that ( 2w ) from the right-hand side. I just want the radical sign. If I subtract ( 2w ) from both sides, what am I left with? Well, on the left-hand side, I am left with ( 6 + 3w - 2w ). Well, ( 3 ) of something take away ( 2 ) of them, you're going to be left with ( w ).

So, ( 6 + w = \sqrt{2w + 12} ).

Now, to get rid of the radical, we're going to square both sides. We've seen before that this process right over here is a little bit tricky because when you're squaring a radical in a radical equation like this and then you solve, you might find an extraneous solution. What do I mean by that? Well, we're going to get the same result whether we square this or whether we square that because when you square a negative, it becomes a positive. But those are fundamentally two different equations.

We only want the solutions that satisfy the one that doesn't have the negative there. So that's why we're going to test our solutions to make sure that they're valid for our original equation.

If we square both sides, on the left-hand side we're going to have ( (6 + w)^2 ). It's going to be ( w^2 + 2(6)(w) + 6^2 ). So, ( w^2 + 12w + 36 ) is equal to ( 2w + 12 ).

Now we can subtract ( 2w ) and ( 12 ) from both sides. So let's do that, so then we can get it into kind of a standard quadratic form.

So let's subtract ( 2w ) from both sides and let's subtract ( 12 ) from both sides. Once again, I just want to get rid of this on the right-hand side. I am going to be left with ( w^2 + (12w - 2w) + (36 - 12) = 0 ).

So, ( w^2 + 10w + 24 = 0 ). Let's see, to solve this, is this factorable? Are there two numbers that add up to ( 10 ) and whose product is ( 24 )? Well, what jumps out at me is ( 6 ) and ( 4 ).

So we can rewrite this as ( (w + 4)(w + 6) = 0 ).

If I have the product of two things equaling zero, to solve this, either one or both of them could be equal to zero. Zero times anything is going to be zero. So, ( w + 4 = 0 ) or ( w + 6 = 0 ).

Over here, if you subtract ( 4 ) from both sides, you get ( w = -4 ) or subtract ( 6 ) from both sides here, ( w = -6 ).

Now, let's verify that these actually are solutions to our original equation. Remember, our original equation was ( 6 + 3w = \sqrt{2w + 12} + 2w ).

So let's see if ( w = -4 ) works.

If ( w = -4 ), that gives us ( 6 + 3(-4) = \sqrt{2(-4) + 12} + 2(-4) ).

So this would be ( 6 - 12 = \sqrt{-8 + 12} - 8 ).

This simplifies to ( -6 = \sqrt{4} - 8 ), or ( -6 = 2 - 8 ).

That indeed holds true, ( -6 = -6 ).

So this is definitely a solution.

Now, let's try ( w = -6 ).

So if ( w = -6 ), we get ( 6 + 3(-6) = \sqrt{2(-6) + 12} + 2(-6) ).

This gives us ( 6 - 18 = \sqrt{-12 + 12} - 12 ).

So we have ( -12 = 0 - 12 ), which is also true.

So we get ( -12 = -12 ).

Therefore, these are actually both solutions to our original radical equation.

More Articles

View All
Power dissipation in resistors in series versus in parallel
A student builds a circuit with a battery and two light bulbs in series. Then the student builds a second circuit with two light bulbs in parallel. Which battery runs out of power first? Assume all bulbs have equal resistance. Assume both batteries have …
How to Apply and Succeed at Y Combinator by Dalton Caldwell
Okay, what we’re going to talk about today is how to apply and succeed at YC. In my capacity as the head of admissions, I go around the world, and I give talks, and I meet folks that are applying, and I answer lots of questions. So I think I have a decent…
The Terlingua Way | Badlands, Texas
If you move here from another place, don’t expect to come out here with 50 bucks in your pocket and a half a dozen 2x4s because you ain’t going to make it. When you come out here, you got to remember we got one cop for a very large area. We have no doctor…
The TOP 5 WORST Credit Cards In 2024
[Music] What’s up guys, it’s Graham here. So normally, on the channel, we talk about the best credit cards to help build your score, give you unlimited cashback rewards, reward you with sign-up bonuses, and do all of that with unparalleled customer servic…
Fentanyl Explained #shorts
Why does fentanyl feel so good? Let us try it so you don’t have to. Fentanyl reaches your brain in seconds, and like other opioids, binds to opioid receptors. It stops pain signals and also releases a flood of dopamine, so the pain melts away as you slide…
This Is A Light-Nanosecond!
I trimmed my beard yesterday, so I’m feeling a bit like a baby today. But look at the trimmings! Specifically this one that is 2.4 fortnits worth of beard growth. How do I know? Well, because of this tool I made. This about 5 years ago to free people from…