The Square Root Property - Video Tutorials & Practice Problems

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Solving Quadratic Equations by the Square Root Property

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6m

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Hey, everyone. So we just learned how to solve quadratic equations by factoring. So if I'm given something like X squared plus X minus six equals zero, then I can just factor that into X plus three times X minus two and get my solutions from there. And that's because this equation is easily factor. But if I'm given something like X squared minus five equals zero, well, I'm not really sure what the factors would be there. And that's actually because this equation is not factor at all. Not all quadratic equations are going to be able to be solved by factoring. But that's OK because there are actually three other methods that we can use to solve quadratic equations. Now, I know that that might sound a little intimidating right now and like a lot, but I'm going to take you through each of these methods step by step and soon you'll be able to solve any quadratic equation that gets thrown at you. So let's go ahead and jump in. So we're going to start filling out this big table of all of the information that you will ever need to know about quadratic equations. And we already looked at our steps to solve a quadratic equation by factoring. But we need to look at another piece of information here, which is when we should use each method because there are multiple. So with factoring, if a quadratic equation had obvious factors or if it has no constant term. So in my standard form equation, if C is equal to zero, then factoring is going to be a good choice to solve it. But if these two things are not true, then we're going to need a new method. So here we're going to look at the square root property. Now, if those two things are not true, but I'm given something in the form of X plus A number squared is equal to a constant. So something like X plus one squared equals four or if I have no X term. So in my standard form equation B is zero and I have something like four X squared minus five equals zero. Then I'm going to go ahead and use the square root property. Now the square root property tells us that we can solve by literally just taking the square root of our quadratic equation. So let's go ahead and see that in action. So looking at my first example, here I have X plus one squared equals four. So looking at our steps, step number one is going to be to isolate our squared expression. So whatever is being squared, I want that by itself. Now, here I have X plus one being squared. So I want that by itself and it already is my squared term is by itself on that left side. So step one is done. Now step two is going to be to take both the positive and the negative square root, this is what's going to give us both of our solutions. So if I go ahead and square root my entire quadratic equation, this left side is going to cancel the square leaving me with X plus one. But then on that right side, I need to take both the positive and the negative square root. So this is going to become X plus one equals plus or minus root four. OK? So we can actually simplify this a little bit more X plus one equal to plus or minus. We know that the root four or the square root of four is just two. So this is really just plus or minus two. OK? So step two is done. Now, we can go ahead and just solve for X for step three. So here if I move my one over to the other side by just subtracting it, I, I leave X by itself and I just have X is equal to plus or minus two minus one. Now that looks a little funky the way it's written now. So if I just rearrange this and move my one over to the front, I will be left with X equals negative one plus or minus two. Now I'm actually going to be able to further simplify this if I split it into my plus and my minus and my answer. So let's go ahead and do that. So this really can become negative one plus two and negative one minus two. Those are my two solutions. Now negative one plus two is just one and then negative one minus two is negative three. So these are actually going to be my two solutions here which I can rewrite a little bit nicer as X equals one and X equals negative three. Those are my final solutions which I saw by just taking the square root. Let's take a look at one more example. So over here, I have four X squared minus five equals zero. Let's go ahead and start back at step one which is to isolate our squared expression. Now here I just have X squared. So I want to get that by itself. Now, the first thing I need to do is move by five over to the other side which I can do by adding five, it will cancel, leaving you with four, X squared is equal to five. Now, one more step to isolate that squared expression is to divide by four, leaving me with X squared is equal to 5/4. OK. So step one is done, we've isolated our squared expression. Now we can go ahead and move to step two, which is to take our positive and negative square root. So if I go ahead and square root, my quadratic equation, it will cancel on this side, leaving me with X is equal to plus or minus the square root of 5/4. Make sure you don't forget that you're taking both the positive and negative square root here. OK. So we can actually simplify this a little bit further and plus or minus root 5/4, I can just split those square roots because I know that I can further simplify the square root of four. So this is really just plus or minus root 5/2. So step two is done, we've taken our positive and negative square root and now we're left to solve for X but X is actually already by itself here. So I'm done. And these are my solutions. Now remember that you can always take your solutions and plug them back into your original equation to check and I leave that up to you. But here we're done. Our solution is just plus or minus route 5/2. I could always split that into the positive and negative solutions, but it's not gonna further simplify it here. So I'll just leave it at that. Now, you might have noticed that our answers looked a little bit different here. So on one hand, I have X equals one and X equals negative three. And over here I have the square root of 5/2. And that's actually because our solutions are not always going to be whole numbers. Sometimes they might have fractions or even radicals in them or some combination of both. And that's totally fine. Your solution doesn't have to be a whole number. So that's all you need to know about the square root property. Let's get some practice.

2

Problem

Problem

Solve the given quadratic equation using the square root property. $\left(x-\frac12\right)^2-5=0$

A

$x=\frac12+\sqrt5,x=\frac12-\sqrt5$

B

$x=\frac52,x=-\frac52$

C

$x=\frac{\sqrt5}{2},x=-\frac{\sqrt5}{2}$

D

$x=\frac12,x=-\frac12$

3

Problem

Problem

Solve the given quadratic equation using the square root property. $2x^2-16=0$

A

$x=0,x=-2$

B

$x=4\sqrt2,x=-4\sqrt2$

C

$x=4,x=-4$

D

$x=2\sqrt2,x=-2\sqrt2$

4

concept

Imaginary Roots with the Square Root Property

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2m

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Hey, everyone, as you solve quadratic equations using the square root property, you may end up getting an imaginary or a complex root. But that's totally fine because we know how to deal with complex numbers. And we're just going to simplify it as we would any complex number. So let's look at an example here, we have four X squared plus 25 equal to zero. And I want to solve this using the square root property. So starting with step one, of course, I want to isolate my squared expression. Now my squared expression is just this X squared. So I want to go ahead and get that by itself by first moving this 25 over. And I can do that by just subtracting 25 from both sides. I'm left with four X squared is equal to negative 25 and then divide by four to get that X squared by itself. I am left with X squared is equal to negative 25/4. So step one is done, I have isolated my squared expression and now I want to take the positive and a negative square root. So taking the square root of both sides here. It will cancel on this side. I'm left with X is equal to plus or minus the square root of negative 25/4. Now, I can go ahead and simplify this further and I can separate it out into plus or minus the square root of negative 25 over the square root of four just using my radical rules. Now, I have a negative under the square root here, but it's totally fine because we've seen this before and we know how to deal with this. So let's go ahead and simplify this. Now, this square root of negative 25 is just going to become five I and then we know that the square root of four is just two. So really, this simplifies all the way into plus or minus five I over two. So we have completed step two, we've taken our positive and negative square root and X is actually already by itself. So step three is done as well. And now I just have my solution. So my solution is X is equal to plus or minus five I over two, you and I can of course, separate these into the positive and negative if I want as well. Now, whenever you're dealing with a complex answer, it's totally fine. It's fine to have the imaginary unit. And a clue that you might end up with a complex answer is if in your standard form equation A and C have the same sign. Then you're always going to end up with a complex answer. That's all there is for imaginary roots. I'll see you in the next video.