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
Evolution | Middle school biology | Khan Academy
[Speaker] How many different species or kinds of birds are there? Take a guess. 100, 1,000, more? Well, biologists have estimated that there are at least 10,000 different species of birds all around the world, and some biologists think that there are ev…
How to organize your life using Notion| My Notion Tour + free templates
Hi, guys! What’s up? It’s me, Ruri. Today, I’m back with another video. So today, I’m gonna show you and explain to you my full Notion setup, what I use, how I use. So that you can start your new school year organized and everything put together, you know…
Is rising inequality necessarily bad
The word inequality, by its very nature, at least sounds a little bit unfair. Obviously, everyone’s not getting the same thing; they’re not getting the same income, or they don’t have the same wealth. But a question needs to be asked: Is this necessarily …
The Ponzi Factor | More than half of Madoff's accounts were WINNERS!
Most people understand that a Ponzi scheme is a scam, but what most people don’t realize is that a Ponzi scheme can also produce a lot of winners. It’s not a scam where everyone loses money; a lot of investors who are involved and unaware of the scam can …
15 Ways To DECLUTTER Your Life
When you were little, remember when your mum used to tell you to tidy your room? Yes, we’re going to remind you of that good advice your mom gave you, but we’re going to take it quite a bit further too. Hey, Aluxers! Watch this video right until the end,…
Price Discrimination: Charge Some People More
Are there any other microeconomic concepts outside of zero marginal cost of replication and scale economies that you think are important for people to understand? I think price discrimination is an important thing to understand. What it means is that you…