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Solving quadratics by taking square roots examples

Video transcript
- [Voiceover] So pause the video and see if you can solve for x here. Figure out which x-values will satisfy this equation. All right, let's work through this. So, the way I'm gonna do this is I'm gonna isolate the x plus three squared on one side and the best way to do that is to add four to both sides. So, adding four to both sides will get rid of this four, subtracting four, this negative four on the left-hand side. And so we're just left with x plus three squared. X plus three squared. And on the right-hand side I'm just gonna have zero plus four. So, x plus three squared is equal to four. And so now, I could take the square root of both sides and, or, another way of thinking about it, if I have something-squared equaling four, I could say that that something needs to either be positive or negative two. So, one way of thinking about it is, I'm saying that x plus three is going to be equal to the plus or minus square root of that four. And hopefully this makes intuitive sense for you. If something-squared is equal to four, that means that the something, that means that this something right over here, is going to be equal to the positive square root of four or the negative square root of four. Or it's gonna be equal to positive or negative two. And so we could write that x plus three could be equal to positive two or x plus three could be equal to negative two. Notice, if x plus three was positive two, two-squared is equal to four. If x plus three was negative two, negative two-squared is equal to four. So, either of these would satisfy our equation. So, if x plus three is equal to two, we could just subtract three from both sides to solve for x and we're left with x is equal to negative one. Or, over here we could subtract three from both sides to solve for x. So, or, x is equal to negative two minus three is negative five. So, those are the two possible solutions and you can verify that. Take these x-values, substitute it back in, and then you can see when you substitute it back in if you substitute x equals negative one, then x plus three is equal to two, two-squared is four, minus four is zero. And when x is equal to negative five, negative five plus three is negative two, squared is positive four, minus four is also equal to zero. So, these are the two possible x-values that satisfy the equation. Now let's do another one that's presented to us in a slightly different way. So, we are told that f of x is equal to x minus two squared minus nine. And then we're asked at what x-values does the graph of y equals f of x intersect the x-axis. So, if I'm just generally talking about some graph, so I'm not necessarily gonna draw that y equals f of x. So if I'm just, so that's our y-axis, this is our x-axis. And so if I just have the graph of some function. If I have the graph of some function that looks something like that. Let's say that the y is equal to some other function, not necessarily this f of x. Y is equal to g of x. The x-values where you intersect, where you intersect the x-axis. Well, in order to intersect the x-axis, y must be equal to zero. So, y is equal to zero there. Notice our y-coordinate at either of those points are going to be equal to zero. And that means that our function is equal to zero. So, figuring out the x-values where the graph of y equals f of x intersects the x-axis, this is equivalent to saying, "For what x-values does f of x equal zero?" So we could just say, "For what x-values does this thing right over here "equal zero?" So, let me just write that down. So we could rewrite this as x, x minus two squared minus nine equals zero. We could add nine to both sides and so we could get x minus two squared is equal to nine. And just like we saw before, that means that x minus two is equal to the positive or negative square root of nine. So, we could say x minus two is equal to positive three or x minus two is equal to negative three. Well, you add two to both sides of this, you get x is equal to five, or x is equal to, if we add two to both sides of this equation, you'll get x is equal to negative one. And you can verify that. If x is equal to five, five minus two is three, squared is nine, minus nine is zero. So, the point five comma zero is going to be on this graph. And also, if x is equal to negative one, negative one minus two, negative three. Squared is positive nine, minus nine is zero. So, also the point negative one comma zero is on this graph. So those are the points where, those are the x-values where the function intersects the x-axis.